control system engineers

Abstract: Many problems in control can be expressed as quantified constraints, such as robust forward and backward reachability, synthesis of controllers, solutions to motion planning problems. In this paper, we develop a method for computing inner and outer approximations of global solutions to very general quantified constraints, generalizing widely the state of the art on similar problems, usually restricted to at most one alternation of quantifiers or to local solutions. By global solutions, we mean characterizing accurately the entire set of solutions, within some prescribed bounds. For this, our approach uses a set inversion technique relying on oracles for set inclusion using local solutions to related quantified reachability problems. We exemplify the approach throughout the paper on different examples related to control problems.

Abstract: Fast inference and strict constraint satisfaction are essential for learning-based decision-making. Existing methods often rely on constraint-specific projections or feasible-set parameterizations, while iterative repair can be costly. We introduce Convex-SKM-Net, a convex constraint-satisfaction layer that forms linearization halfspaces from residual and (sub)gradient oracles without constraint-specific inequality projectors. It applies fixed-depth dual projected-gradient descent (dual PGD) to minimum-displacement block problems using GPU-friendly matrix operations. We jointly learn the predictor and a positive-definite mobility, adapting the correction geometry to improve objective quality. We prove convergence of the outer iterates at fixed dual depth and linear convergence in distance under Slater's condition and bounded subgradients. All our test outputs on three second-order-cone model-predictive control (SOC-MPC) horizons and six power-system convex relaxations satisfy a $10^{-3}$ feasibility tolerance. Comparisons with the latest constraint-satisfaction methods show inference speedups of up to $144.3\times$ on SOC-MPC and $7.7\times$ on a convex quadratically constrained quadratic programming (QCQP) benchmark with joint training. On QCQP, the mean objective gap is $0.012\%$, with full test feasibility at $10^{-4}$. Learning mobility reduces the five-seed affine-task mean gap by $22.4\%$ and improves gaps in matched power-system comparisons.

Abstract: In dynamical systems, feedback can improve performance by allowing decisions to depend on past observations, rather than being designed a priori. Under full observation, the current state is sufficient for decision-making; under partial observation, the controller must instead estimate the state from measurements. When actions also affect observations, control can have a dual role: actions influence both the state and the information available for future decisions. This interaction can invalidate certainty-equivalent (CE) control, a widely used strategy that applies state feedback policies as if the estimated state were the true state. We ask when this suboptimality leads to a quantitative performance gap that grows with the horizon. We study this question in a linear-quadratic system with bilinear observations by comparing the optimal policy, certainty-equivalent control, and the optimal open-loop (OL) policy. We identify an instance-dependent information-strength parameter $q$ and establish distinct performance regimes as $q$ varies. When $q$ is sufficiently small, we show that no policy can improve over OL control by more than $O(1)$ cumulative cost. For large values of $q$, we find that CE achieves an $Ω(T)$ advantage over OL control. Our main finding is that for intermediate values of $q$, we can construct a policy that occasionally probes for information and achieves an $Ω(T)$ advantage over OL control while CE only achieves an $O(1)$ advantage, establishing an $Ω(T)$ gap between CE and optimal control. Our analysis relies on a regret decomposition that captures the tradeoff between the immediate cost of gathering information and its future benefit.

Abstract: We propose a new regression method named Direct Inference on Admissible Sets (Diona) that directly uses samples to estimate the state and its derivatives at a query point, along with their admissible bounds, without long global training. By casting the values and derivatives at the query and neighbouring samples as latent variables in a constrained optimisation problem, the burden of long training loops becomes one of on-demand local solves. We establish Diona's theoretical properties including finite-sample coverage and, in the noiseless case, convergence to the true state and derivatives under dense sampling. In a nonlinear regression benchmark on six functions of three to eight dimensions, Diona was more accurate than a Gaussian process (GP) on noiseless data especially with denser data sets. Furthermore, as with GP, it produced uncertainty bounds that contained the true value at more than 95% of query points in four functions. Diona's time to a first prediction after a data set update was three to four orders of magnitude shorter than that of the GP, and two to three orders shorter than that of a neural network, which was more accurate but offers no uncertainty quantification. This low latency is paid for by on-demand predictions, so this latency advantage applies when relatively few predictions are needed between data updates.

Abstract: Continuous-time stochastic differential equations (SDEs) provide a natural framework for modeling interacting dynamical systems. A central causal question is whether causal effects among observed variables can be identified from observational data drawn from the stationary distribution, given the underlying causal graph. For linear SDEs, existing identifiability results are largely restricted to fully observed systems whose causal graphs contain no directed 2-cycles. We first establish new graphical conditions for the generic identifiability of causal effects in systems that may contain directed 2-cycles, thereby allowing for reciprocal interactions. Next, we address the previously unexplored setting of latent variables by developing an instrumental-variable framework for identifying a target direct causal effect under partial observability.

Abstract: Trained network weights are increasingly treated as data, but there is rarely ground truth about what a population of models should encode, which hampers systematic validation of weight-space methods. This makes parametric dynamical systems a natural test-bed: the dynamics change qualitatively at submanifolds of parameter space that are, in well-studied families, known analytically or numerically. Fine-tuning a forecasting network at each point of a parameter grid yields a model zoo indexed by a physical parameter, against which weight-space diagnostics can be checked. We build such model zoos for four dynamical systems and analyse them with JEPAWG, a recently proposed joint-embedding hypernetwork built around a learned parameter--weight representation. It was first introduced for neural samplers for lattice field theories, where its latent geometry located the phase transition and estimated the 2D Ising critical exponent. Applied to dynamical systems, JEPAWG is competitive on forecasting with a hypernetwork trained directly for that task, despite being supervised on the parameter--weight correspondence rather than on forecasts. Beyond forecasting, the norm of the parameter-derivative of the decoded weights locates Hopf points and curves, tracks a period-doubling cascade closely enough to give a qualitative estimate of the Feigenbaum constant, and detects a global bifurcation, which has no eigenvalue signature and is therefore invisible to linearisation. As such, JEPAWG provides interpretable insight into the dynamics that is not available from other hypernetworks. Together, these results indicate that JEPAWG is a workable instrument for reading physical structure off a population of trained networks.

Abstract: We study worst-case finite-time energy dissipation in quadratic systems with proved global asymptotic stability (GAS). Estimating the maximum retained energy fraction to any fixed positive additive error below $1/2$ is NP-hard, even when the input includes a strict quartic Lyapunov function and proof of strictness, both constructed in polynomial time. NP denotes nondeterministic polynomial time. For a partially damped family, the unit-sphere minimum norm of the coupling map equals the structured coefficient distance to loss of GAS. Positive distances at most $1/2$ give two-sided inverse-square halving-time bounds with explicit coefficient factors. Polynomial-factor approximation of the least dissipated fraction, halving time or coefficient distance is NP-hard. Additive hardness persists throughout an explicit coefficient neighborhood and under direct damping with exponential stability, a strict quadratic Lyapunov function and a Hurwitz Jacobian. Unless $\mathrm{P}=\mathrm{NP}$, where P denotes deterministic polynomial time, no polynomial-time algorithm attains these guarantees on all inputs.

Abstract: In this paper, we study continuous-time reinforcement learning, where the state follows an unknown stochastic differential equation but only a fixed off-policy batch of discrete-time transitions is available. Standard discrete-time Q-learning becomes problematic as the sampling interval shrinks, because differences between action values vanish. Continuous-time alternatives, such as Deep Advantage Updating and martingale q-learning, preserve the action signal but jointly learn a value function and an action-dependent critic. Building on the PhiBE-Q formulation (arXiv:2606.21925), we develop a model-free PhiBE policy iteration (PhiBE-PI) framework that uses a single state-action critic while preserving the continuous-time action-ranking scale. At each iteration, the framework learns the critic directly from sampled increments and uses it to improve the policy. The same batch is reused throughout, without explicitly estimating the drift or diffusion. We further develop an implementation under linear function approximation and establish a local end-to-end suboptimality guarantee for the learned policy. Our analysis quantifies how errors from time discretization, function approximation, and finite-sample policy evaluation propagate through policy iteration. It accounts for the dependence of successive policies on the reused data, so the guarantee does not require a fresh batch at each iteration. Moreover, we identify a bias-variance trade-off: smaller sampling intervals improve the approximation to the continuous-time problem but can increase statistical error at a fixed sample size. Thus, finer time discretization does not necessarily lead to better policy performance.

Abstract: While statistical learning has made remarkable progress in recent decades, driven by large datasets and data-driven protocols that have transformed scientific practice, it still faces a key limitation: it struggles to generalize beyond the data it has seen, often failing under distribution shift. Causality offers a promising solution: causal models focus on the underlying mechanisms of change rather than surface-level associations, enabling robust reasoning under external interventions. This allows systems not only to learn from data, but also to reason about hypothetical scenarios, adapt to new environments, and inform better decisions. This thesis explores how these approaches can inform each other, in two parts. The first part examines algorithmic recourse, that is, identifying which changes an individual can make to overturn an unfavorable decision. We propose an alternative perspective to the established framework, arguing for a more realistic cost function, a reformulation of the problem in causal terms, and the inclusion of a temporal dimension in policy evaluation. The second part focuses on dynamical systems, extending the notions of causal interventions and effects to discrete and continuous stochastic processes. We show that the equilibria of these processes, and how they change following an intervention, can be linked to Structural Causal Models (SCMs), viewed as compressed representations of the underlying temporal dynamics. We argue for the relevance of such temporal causal models, since in real-world scenarios actions take time to produce their effects. The common goal of both parts is practical: the utility, robustness, and trustworthiness of these models depend critically on their ability to realistically represent the underlying phenomena.

Abstract: This article presents an axiomatization of linear differential equations that is complete for topologically open properties with compact initial conditions and time horizons by proving them deductively via a sound reduction to formulas of linear real arithmetic. Indeed, linearity is preserved: linear differential invariants are complete for open properties of linear differential equations and reduce with a linear number of Lie derivatives (per atom) to linear real arithmetic. In particular, this article shows that the linear fragment of differential dynamic logic (dL), a modal extension of standard first-order logic over the reals satisfies these completeness properties. By establishing completeness within decidable linear real arithmetic, the need for proving sophisticated nonlinear formulas of real arithmetic is circumvented. Furthermore, this article establishes generalizations to the nonlinear fragment of dL that enable the practical certification of reachability computations, validating the output of external computations in dL with machine-checkable proofs.

Abstract: If a regular linear differential-algebraic equation has a unique solution, this solution is known to be Turing computable. In contrast, this paper shows that singular linear differential-algebraic equations exhibit fundamentally different computability properties. More precisely, we show that for every linear differential-algebraic equation (LDAE) with constant computable coefficients, there exists a computable input function with strong regularity properties such that the LDAE admits a unique solution, yet this solution is not Turing computable. In fact, the solution is non-computable at certain computable times.

Abstract: Reliable quadrotor flight in disturbed environments requires predictive models that remain accurate as unmodeled aerodynamic effects change during deployment. We introduce a disturbance-aware Koopman Control Family (DA-KCF) framework that preserves an offline-identified nominal model while learning only a disturbance-induced correction online. A reference-dependent KCF is first learned from simulation data and calibrated using limited hardware data. During deployment, the nominal KCF remains fixed while a lightweight residual ensemble learns a low-dimensional disturbance correction from causal state-command-state transitions. The Adaptive DA-KCF model is incorporated into MPC, where disturbance corrections are evaluated along the prediction horizon to optimize position and yaw references sent to the onboard stabilizing controller. Hardware experiments on a nano-quadrotor under uniform wind and a spatially varying wake demonstrate that online adaptation improves multi-step prediction and trajectory tracking as disturbance-induced model mismatch increases, while satisfying real-time control requirements.

Abstract: Physics-informed neural networks (PINNs) and other residual-trained solvers judge a candidate solution by the size of its sampled PDE residual. For first-order Hamilton-Jacobi equations this is not sufficient: the same equation and boundary data admit Lipschitz almost-everywhere solutions that differ from the viscosity solution by $O(1)$ while violating the equation only on sets of measure zero. We study whether residual minimization selects such solutions in a fixed-horizon pursuit-evasion game whose Hamilton-Jacobi-Isaacs value function is known in closed form on the circle $S^1$ and on the flat tori $\mathbb{T}^{2}$ and $\mathbb{T}^{4}$. We train four neural parameterizations with ten seeds on each geometry, 120 runs in total. A standard network with periodic Fourier features converges to folded almost-everywhere solutions in all 30 runs: small residuals and the expected capture-layer gradient, yet values that are wrong over most of the domain. A network with periodicity imposed through the loss recovers the viscosity solution in one and two dimensions but becomes unreliable in four. Two parameterizations that receive the geodesic distance do substantially better: a single network fed the wrapped displacement and its distance recovers the viscosity solution in all 30 runs, while a minimum over periodic image charts succeeds in 27 of 30 and introduces a wrong-image failure mode. Finally, for the convex Hamiltonian considered here, we use the semiconcavity of the viscosity solution to build a mesh-free finite-difference test that needs only forward network evaluations; it detects the folded solutions and the harmful min-of-charts failures without access to the exact value function. A small sampled residual thus does not by itself validate a learned Hamilton-Jacobi solution; the parameterization and independent structural diagnostics can be essential for selecting the viscosity solution.

Abstract: Dynamical systems are a widely used model of computation and an intuitive representation of many natural phenomena in many applied fields. The last decades have seen an impressive use of computers in studying and analysing them, with several visible theoretical breakthroughs. The question of reachability for dynamical systems is central: given two points, is there a trajectory between them in the dynamical system? Reachability, in particular over the reals, is undecidable in the general case: it is only computably enumerable, even under reasonable hypotheses, e.g. for a computable dynamics in general. But it is known to become decidable, and even lies in PSPACE, under suitably quantified robustness assumptions. In this paper, we focus on PSPACE-completeness results over the reals, in the sense of computable analysis, regarding discrete-time and continuous-time robust dynamical systems. We prove that, under some additional and rather reasonable hypotheses, the reachability relation becomes even PSPACE-complete for both classical and second-order complexity second-order complexity, for an adapted notion of reduction. As a side-effect, we also prove an improvement of an algebraic Ordinary Differential Equations-based characterisation of PSPACE, removing one of the base functions of the algebra. By doing so, we link results from dynamical systems and complexity over the reals, including the models of computable analysis and second-order complexity.

Abstract: To procure a model of a device, a client hands a supplier a description, sometimes with recorded data or a target response, and accepts the model against this handover. We let LLM coding agents act as the supplier on handovers with data, building Modelica models from library components for four simulated systems and a real switching magnet, and ask what acceptance certifies. Several agents reach the noise floor, but a handover admits a set of models: the description leaves parts of the structure open, and the data narrow it only to the agreed accuracy. We therefore accept by membership in this set, not by matching one reference, and add layers above behaviour: a physical invariant exposes an energy-creating model at the noise floor. Interpretability becomes a property of the delivered model, tested on a variant, read where none exists and shaped through the handover. Most failures are fits delivered with a systematic residual on their own fitting data, where the handover left open how close is close enough. In a follow-up, stating a threshold makes the agents compare against it, and more deliveries pass. The argument applies wherever a handover underspecifies a model meant to be read, not only run.

Abstract: We revisit smoothed online learning and develop algorithms with improved competitive ratios, optimal dynamic regret with switching cost, and best-of-both-worlds guarantees. Surprisingly, minimizing the hitting cost alone is $\max(1,\frac{2}α)$-competitive for $α$-polyhedral functions under the $\ell_2$-norm switching cost and $1+\frac{4}λ$-competitive for $λ$-quadratic growth functions under the squared $\ell_2$-norm switching cost, significantly improving existing guarantees. The polyhedral bound is tight for this naive algorithm, and we establish a matching lower bound for $α\geq1$. For convex $λ$-quadratic growth hitting costs, minimizing a suitably weighted sum of hitting and squared $\ell_2$-norm switching costs yields a competitive ratio of $1+\frac{2}{\sqrtλ}$. For dynamic regret with the $\ell_2$-norm switching cost, we develop Smoothed Ader (SAder), which achieves the optimal $O(\sqrt{T(1+P_T)})$ rate in the standard online convex optimization setting, where $P_T$ is the comparator's path-length. Its lookahead version removes the bounded-gradient assumption, and a matching lower bound establishes optimality in this setting. Finally, for convex hitting costs on a bounded convex domain, we develop UDNP to aggregate any two experts in the lookahead setting under both switching costs. Its additional cumulative cost is at most $O(1)$ relative to the first expert and $O(\sqrt{T\log T})$ relative to the second, yielding best-of-both-worlds guarantees without bounded gradients.

Abstract: Gaussian process distributed port-Hamiltonian systems (GP-dPHS) enable data-driven modeling of distributed dynamical systems while preserving their physical structure. However, standard stationary kernels can be inefficient for spatially heterogeneous systems because they neither exploit known input invariances nor reflect structured spatial dependencies. We address this limitation with a structure-aware kernel design that removes irrelevant state coordinates and decouples state and spatial dependencies through a tensor-product covariance. This modular construction allows qualitative knowledge of spatial heterogeneity to be incorporated by selecting an appropriate spatial kernel without requiring an analytical description of the underlying profile. Numerical experiments on a nonlinear string with periodic, smoothly varying, and semi-piecewise stiffness profiles demonstrate that structural decoupling consistently improves reconstruction accuracy, while profile-matched spatial kernels yield substantially larger gains in force-field and trajectory reconstruction.

Abstract: Neural networks offer critical adaptability for controlling complex systems where nominal modeling proves intractable. However, when offline-trained neural networks are deployed within closed-loop control systems, their actions fundamentally alter the state distribution upon which they were trained. This paper introduces a unified taxonomy to categorize these sequential failure modes: the initial Closed-Loop Identification Error, the deployment-induced Action Error, and the continuous Tracking Error. Through discrete-time sensitivity analysis, this work formalizes how offline-learned parameters locally guarantee a trajectory divergence upon deployment, degrading performance due to local loss geometry. To mitigate this distribution shift, we propose actionable design policies centered on time-scale separated online adaptation. These mechanisms are validated through ablation studies on a nonlinear Duffing oscillator, demonstrating that bounded continuous learning successfully mitigates trajectory divergence whereas static deployments fail, offering concrete guidelines for practitioners deploying machine learning components in safety-critical feedback systems.

Abstract: This paper proposes the SKG (Symplectic-Kernel Gradient) reasoning engine for law-induced inference and induction. Laws are a priori; data provide instances. Inference is geometric evolution under law constraints. We organize symplectic geometry, GENERIC, kernel methods, RKHS, group-equivariant networks, and constrained mechanics into a unified, analyzable, implementable, verifiable framework. We propose the SKG octuple S = (M, g, omega, H, S, G, Delta, Pi), whose four components correspond to four levels of laws: conservation laws, constitutive relations, symmetry groups, and axiom systems. Main results: (1) Invariants enter metric construction via the Jacobian: g = J^T Sigma^-1 J + epsilon_G I. For constant cases, we establish inverse reconstruction theory: solvability rank condition (Thm 3.9), consistency criterion (Thm 3.11), and uniqueness theorem (Thm 3.14). (2) The kernel bandwidth matrix is uniquely determined by Fisher information and satisfies task-independence (Thm 3.19). (3) The metric reconstruction error decomposes into four terms (Thm 3.21), with squared condition number kappa(Sigma^*)^2 as the dominant amplification factor (Thm 3.22). (4) For state-dependent cases, we establish function-space reconstruction theory on matrix-valued RKHS (Thms 4.8, 4.9). (5) Three-layer projections handle axiom, symplectic-step, and conservation constraints. (6) The residual equation yields the observable formula, four-dimensional rank condition, and noise propagation, while g_qq is structurally unidentifiable. (7) Kernel parameterization yields the RKHS extrapolation bound (Thm 4.15) and strict OOD gain theorem in group directions (Thm 8.15). (8) The learning-decoupling theorem (Thm 9.1) proves the sole object of induction is the parameters of the task energy. Validation: 1D damped oscillator and 2D rotation-invariant system; 6 unit tests, 10 experiments (paired t-test, p < 0.01).

Abstract: We study learned one-step predictive control of a synthetic population of 64 leaky integrate-and-fire neurons whose light sensitivity is scaled by a hidden factor s, using two prospective confirmations on plants simulated only after the analysis was frozen. In the first (50 plants, 11 conditions), removing the 200-ms input-output history substantially worsened a parameter-count-matched neural predictor at low and moderate gain (s = 0.6, 0.8; RMSE rose by 2.47 Hz at s = 0.6, 97.5% CI 2.43-2.51). In secondary comparisons, recursively adapted ARX had the lowest mean error of the deployable controllers in 10 of 11 conditions, tuned PI with anti-windup matched or beat the network for s >= 0.7, and training for 10,000 rather than 900 epochs lowered prediction error but worsened control. Development work suggested the network's learned action sensitivity did not track the hidden gain. A second protocol (50 new plants, 9 sensitivities, 2 noise levels) confirmed three pre-specified predictions at the primary noise level. Re-estimating a single multiplicative input gain recovered 87% (95% CI 86-88%) of the benefit of full adaptive ARX. Supplying that online estimate to the network as an extra input reduced high-gain tracking error by 1.84 Hz [1.81, 1.88]. Longer training worsened the plain network by 1.09 Hz but improved the gain-input network by 1.07 Hz, a differential of 2.16 Hz [2.13, 2.20]. Under high process noise (sigma = 2.0) the effects shrank to about 0.04 Hz or less and gain-only adaptation recovered only 24% of the adaptive-ARX benefit. In this synthetic plant, at the primary noise level, one-parameter online gain estimation accounts for most of the gap between a history-based network and full adaptive ARX, and for its prediction-control mismatch.

Abstract: Timely information dissemination is essential in modern communication systems, where low informational latency is critical for networked control, monitoring, and real-time decision-making. The Age of Information (AoI) has emerged as a fundamental metric for quantifying information freshness at the receiver. A key challenge in AoI optimization is selecting the update generation rate: infrequent updates lead to stale information, whereas overly frequent updates increase queueing delays. We develop a structure-aware, data-driven framework for adaptive AoI optimization modeled as single-server queues with unknown system statistics. Our key idea is to exploit analytical representations of the AoI gradient, which depend on a finite-dimensional collection of system quantities that can be estimated from observations collected during system operation. This reduces the online learning problem to estimating the quantities required by the gradient, rather than learning the AoI objective as a black box. We propose an online stochastic approximation algorithm that jointly estimates these quantities and optimizes the update generation rate. We prove almost sure convergence of the algorithm to the optimal update generation rate under mild regularity and moment conditions, despite the temporal dependence induced by the underlying queueing process. We demonstrate the framework in four representative incomplete-information settings, including FCFS queues with unknown service characteristics or hidden background traffic, a preemptive LCFS system with partial observations, and a generate-at-will system with threshold waiting.

Abstract: We consider the problem of learning linear dynamical systems under adversarial contamination from a single trajectory of length $T$. While identification of linear dynamical systems itself is well-studied, the problem of robust system identification under adversarial contamination is relatively less explored. In this work, we study the setting where a fraction of the $T$ observations are contaminated by adversarial outliers. We propose different estimators based on relaxations of least-trimmed squares along with an alternating minimization algorithm. Furthermore, we also propose two estimators which exploit the group-sparsity (through penalization/hard-constraints) of the outliers. For the estimator with group-sparse penalty, we derive non-asymptotic error bounds which establish its robustness to outliers. We also show empirically that the proposed estimators work well in practice.

Abstract: Policy learning drives many of the most consequential and heavily-invested applications of reinforcement learning today. Yet the core optimization problem it rests on (maximizing expected return) is notoriously non-convex, even under a direct policy parameterization, and the field has largely responded by avoiding it: optimizing convex surrogate approximations of the return under trust-region constraints (NPG, TRPO, PPO, AWR). We show that this seemingly unstructured problem is not actually structureless. In log-density-ratio coordinates $y := \log[π/π_n]$, the exact per-iteration objective, computable via per-decision importance sampling (PDIS), is a difference-of-convex-constrained difference-of-convex (DC-constrained DC) program. This structure lets us move beyond surrogate approximations: it recovers CPI, NPG, TRPO, and AWR as special cases along interpretable axes, and it opens a multi-step axis $k$ that couples consecutive decisions. We solve the per-iteration program with sequential convex programming (SCP), the standard solver for difference-of-convex problems, and give convergence guarantees under mild conditions, bridging the difference-of-convex optimization and RL literatures. Empirically, multi-step Convex-Concave RL (CCRL) wins on diagnostic MDPs where credit must propagate across a horizon (its advantage growing with the dependency length), is competitive with a tuned PPO on classic control, and on a realistic, stochastic, mid-horizon healthcare domain converges markedly faster than tuned PPO to the same near-optimal survival, with an 11.3% higher area under the training curve.

Abstract: Covariate-aware time-series foundation models (TSFMs) promise training-free what-if answers for instrumented plants: the change in output that a different future input would cause. We test this on forced engineering systems with exact counterfactuals, comparing Chronos-2, TimesFM-2.5 and TabPFN-TS with classical system identification fitted to the same context. Through their default covariate interfaces, TimesFM-2.5 and TabPFN-TS are memoryless: the predicted effect of an input change is a same-time function of that change ($R^2 = 1.000$ for TimesFM-2.5). Chronos-2 identifies dynamics in context but attenuates them. Its predicted effect is 0.33-0.80 of the true effect, its recovered impulse response has the wrong shape, and its error on a one-degree-of-freedom oscillator levels off at 0.57 with 8192 context samples, where ARX fitted to 256 samples reaches 0.02. Context dither at inference lowers the what-if error on all six synthetic classes without training. A 26-minute fine-tune on synthetic forced systems restores the response magnitude (sensitivity 0.83-0.96) and outperforms structure-agnostic identification on Wiener-Hammerstein and a held-out friction class. A specialised in-context identifier trained on the same data comes close, so the forced-system data carry most of the gain. On three of four measured plants classical identification remains clearly better, and the fine-tuned model loses part of its univariate forecasting skill. Paired counterfactual inputs, together with shuffled future inputs on measured records, test two properties: whether the covariate interface can represent dynamics and whether the pretraining prior covers the plant's time scale. Only the counterfactual pairs expose the attenuation.

Abstract: Reservoir computing (RC) is a machine learning technique for data-driven modeling of dynamics for forecasting and control, primarily studied in computer science and physics literature with promising applications in neural network learning, physical computing, and neuroscience. Why reservoirs learn and how to choose good reservoir architectures are considered important open questions. We show that the RC problem is mathematically an extension of a classic problem in systems and control theory, the Kazantzis-Kravaris-Luenberger (KKL) observer design problem. As a consequence, many of the questions considered open for RC stand to benefit from a large body of theory in the mature KKL literature, non-exhaustively including on questions of embedding, transverse stability, local and global uniqueness guarantees, and effective data-driven solution constructions. Elaborating on this connection, we show that the surprising forecasting ability of reservoirs is in fact a direct consequence of the well-known observer internal model principle, derive an upper bound on the prediction error over a fixed forecast horizon, and provide a partial explanation for why linear readout training in RC works reasonably well. This work illustrates how classical ideas from systems and control can provide strong theoretical backing and open new questions for modern machine learning methods.

Abstract: Understanding a nonlinear dynamical system from time series requires not only reproducing its trajectories, but also identifying a simple representation that preserves its essential dynamical structure. Almost-linear recurrent neural networks (AL-RNNs) are piecewise-linear RNNs in which only a subset of units use ReLU nonlinearities, so that nonlinear capacity is explicitly controlled by the number of ReLU units. Their activation patterns define linear regions, represented as symbols, whose observed transitions form a symbolic transition graph. However, directly training AL-RNNs with few ReLU units to realize minimal dynamical representations can be unreliable. We ask whether an AL-RNN with more ReLU units can instead be trained first and systematically reduced to a minimal dynamical representation. We introduce a fixed-point-guided hierarchical reduction procedure that progressively linearizes selected ReLU units, merging neighboring linear regions and graph nodes while preserving distinct symbols containing fixed points (FPs). The resulting reduction tree defines a hierarchy of progressively simpler candidates. Each reduced candidate is initialized from the parent parameters and retrained under guidance from the parent dynamics. We also prove that reproducing $Q$ distinct fixed points requires at least $Q$ FP-containing symbols, providing a certificate of symbol-level minimality when this bound is attained. On the 3-scroll Chua system, direct training with the theoretical minimum of three ReLU units achieves high-fidelity minimal realizations in only 20% of seeds, whereas our learn-reduce-retrain strategy increases the seed-macro success rate to approximately 71% at the same final nonlinear capacity. These results show that redundant nonlinear capacity can serve as a scaffold for discovering and realizing minimal dynamical representations.

Abstract: A stochastic hybrid system (SHS) is governed by a stochastic differential equation (SDE) describing the continuous dynamics and a Markov reset kernel triggered on the guard surface. Its probability evolution can be described by a hybrid Fokker-Planck (HFP) equation with a partial differential term corresponding to the SDE and an integral term arising from the reset kernel. This work shows that such an SHS can be approximated by an SDE in a higher-dimensional latent space where the sample paths are continuous. The key to this result is to encode different branches of the reset kernel using auxiliary variables, transforming the resets into deterministic ones that enable topological gluing. By the embedding theorem, the glued manifold can then be embedded into a higher-dimensional Euclidean space. We show that the probability evolution on the embedded image no longer requires explicit reset terms in the HFP equation. Building on this theorem, we design a loss that matches the evolving state distributions, enabling a single latent SDE to recover the probability evolution of the SHS without mode labeling, trajectory segmentation, or event-based simulations.

Abstract: Autoregressive models are trained to predict a system's behavior one step at a time, and recursive generation allows the learned dynamics to unfold over long horizons. To what extent can such dynamics learned from local observations recover broader organization of an underlying system that was only partially observed during training? Here we study small autoregressive transformers trained from scratch on trajectories sampled from restricted parameter regimes of several non-linear dynamical systems, including logistic and sine maps, the Lorenz system, and the generalized Hopf system, with control parameters and state trajectories represented as sequences of continuous tokens. Under closed-loop evaluation at parameters far outside the training distribution, the models can recover self-similar period-doubling cascades, chaotic dynamics, and attractor structures with remarkable visual and numerical fidelity. For the logistic map, a transformer reproduces successive period doublings up to period 128, yielding a finite-order scaling ratio of 4.6687, matching the Feigenbaum constant to within $5\times10^{-4}$. We further investigate how these structures emerge over the course of training, and reveal with causal interventions how control-parameter information is processed through attention into state prediction and shapes the resulting closed-loop dynamics. These results suggest that a surprisingly narrow window into a system's local behavior may suffice for autoregressive transformers to generalize to its unseen global dynamical organization.

Abstract: A wide range of methods have been proposed, including physics-informed neural networks, which are powerful but do not guarantee identifiability of the dynamics, symbolic regression, which requires a set of precomputed operations, and causal discovery, which is more principled but usually relies on strong assumptions that physical systems may violate. In this work, we develop a theory-grounded method and prove that under a set of permissive assumptions, the structural drivers and drift of stochastic delayed differential equations are identifiable. Our method outperforms others on a benchmark for driver identifiability, and on a second benchmark to evaluate physical consistency of the learned dynamics.

Abstract: We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing methods for ODE discovery typically assume direct measurements of the variables, or do not provide theoretical guarantees on the learned variables and equations. While Causal Representation Learning (CRL) methods provide guarantees on identifying variables from high-dimensional observations up to component-wise diffeomorphisms, we show that in general these variables cannot be used directly as input to equation discovery methods, which typically assume that the variables will lead to sparse equations. So we introduce SParse Equivalent Equation Discovery AutoEncoder (SPEED-AE), a framework that combines a pretrained CRL method with a component-wise autoencoder that learns transformations of variables that are amenable to sparse ODE discovery. We show that for polynomial ODEs, this additional step allows us to restrict the identifiability of each variable from polynomial to monomial diffeomorphisms. Experiments on Lotka-Volterra, Lorenz, and a two-pendulum system show that SPEED-AE improves on the disentanglement of the CRL methods and that it recovers ODEs that are closest to the ground truth, while achieving state-of-the-art forecasting performance.

Abstract: Accurately learning nonlinear dynamics from a finite-duration experiment requires the efficient collection of informative data. We address this challenge for stochastic controlled nonlinear dynamical systems whose state is observed along a single trajectory. Our goal is to reconstruct the unknown controlled state-increment map over a prescribed compact subset of state-input space. We construct a parametric estimator of the map using fixed nonlinear features, so that the model is nonlinear in the state and input, but linear in the unknown parameters. A Gaussian prior over the parameters yields recursive Bayesian posterior updates as data stream in, enabling online quantification of predictive uncertainty in the reconstructed dynamics over the target set. We formulate an optimal adaptive-design problem over an information state, using a prediction-oriented acquisition criterion based on the mean marginal mutual information between candidate future trajectories and the reconstructed dynamics over the target set. We then approximate the resulting adaptive-design problem by a non-myopic receding-horizon formulation, evaluate its remaining expectation using a scenario-based sample average, and solve the resulting deterministic program with the cross-entropy method, leveraging parallel candidate-scenario evaluations. Numerical experiments on a noisy multistable system demonstrate that the proposed adaptive information-seeking strategy reduces predictive uncertainty and reconstruction error more efficiently than common excitation baselines under comparable experimental constraints.

Abstract: Gaussian process stochastic differential equations (GP-SDEs) provide a flexible Bayesian model for unknown continuous-time state dynamics with uncertainty quantification, but learning and inference from noisy and irregular observations remain computationally challenging. To address this issue, we propose GP-SDE Matching, a simulation-free variational framework for Bayesian GP drift learning and continuous-time state smoothing. We analytically marginalize the sparse GP posterior to derive a tractable drift-matching objective that accounts for both the posterior mean and uncertainty of the unknown drift. To handle irregular observations, we further introduce an irregular-time-aware variational state posterior that incorporates the actual observation times during both encoding and continuous-time marginal querying. Experiments on the stochastic Lorenz--63 system demonstrate substantially improved drift recovery and state reconstruction under irregular observations, while five system identification benchmarks show robust forecasting under increasing observation sparsity and competitive performance against existing latent-SDE and state-space methods.

Abstract: Tree-Adjoining Grammars (TAGs) have recently been introduced to Nonlinear System Identification (NLSI) as a means of encoding an entire model class as a finite set of grammatical rules, from which candidate models are assembled as trees. Existing TAG-based identifiers rely on evolutionary optimisation and return point estimates of the model structure. This paper instead proposes the TAG framework within a Bayesian setting. A generative prior is defined over tree structures and their parameters, and a Reversible-Jump MCMC sampler with structure-preserving tree moves is used to infer the joint posterior over model structure, parameters and predictions. Two training objectives are considered; that is, a one-step-ahead objective with conjugate parameter proposals, and a simulation-based objective handled by likelihood-free inference. The approach is validated on a simulated polynomial NARX system, the Silverbox benchmark, and wave-loading data from the Christchurch Bay Tower, where embedding Morison's equation as a fixed initial tree yields a grey-box model that outperforms the physics-driven baseline. The results demonstrate that Bayesian TAGs are well suited to quantifying uncertainty in equation discovery for dynamical systems and to fitting physics-informed models.

Abstract: The minimum cycle mean problem (MCM) on weighted digraphs is the problem of finding the minimum value of the cycle mean, that is, the ratio of the cost to the length, over all cycles. Despite its wide range of applications to discrete event systems, the parametric counterpart of the MCM has received relatively little attention in the literature, unlike other parametric problems in network optimization. In this paper, we consider the linear parametric MCM, where all edges $e$ have cost $a(e)-b(e)t$ with parameter $t$. We propose an algorithm to solve the linear parametric MCM in $O((m+n\log n)n^2W)$ time, where $n$ is the number of vertices, $m$ is the number of edges, and $W$ is the maximum absolute value of the coefficients $b(e) \in \mathbb{Z}$. The central technique of the proposed method is the algorithm for the parametric shortest path problem. The MCM is closely related to spectral theory in the tropical semiring, where the ``$\min$'' operation is regarded as addition and ``$+$'' as multiplication. By exploiting the connection between them, we provide a method to compute the eigenvalues and eigenvectors of tropical parametric matrices.

Abstract: Robust average-reward Markov decision processes provide a fundamental framework for long-term performance optimization under uncertainty, and can have optimal long-run rewards that depend on the initial state. This state dependence requires a vector Bellman theory that accounts for both recurrent-class rewards and transition uncertainty. We develop such a theory for finite models with compact, post-action $(s,a)$-rectangular ambiguity. A gain-first, bias-second optimization principle yields a coupled vector gain-bias system, and every finite solution identifies the optimal robust gain and supplies stationary saddle strategies against history-dependent opponents, simultaneously from all initial states. We further characterize solvability through stationary gain conditions and a uniform bound on canonical transient corrections, and give sufficient conditions that permit distinct recurrent-class gains. The certificates also yield asymptotically affine trajectories of the robust Bellman operator, based on which we design a robust approximately shifted Halpern planning algorithm. Under finite Bellman solvability, the gain estimates and Bellman displacements converge to the optimal gain vector, and every extracted greedy controller is average-optimal after a finite, instance-dependent budget. These results thus connect finite Bellman certificates to undiscounted planning for state-dependent robust average rewards, providing theoretical understandings.

Abstract: Adaptive regulation can itself perturb the state it is intended to stabilize. In replicated simulations of an adaptive agent, we compare external disturbance, persistent internally generated disturbance, and control-generated disturbance under regulation-first and disturbance-first ordering. Persistent internal disturbance produces the largest exposure and regulatory burden within the tested parameter grid. When positive controller updates generate an immediate disturbance cost, increasing that cost produces a nonmonotonic response: effective disturbance initially rises, variability across stochastic runs increases over an intermediate range, and corrective activity becomes strongly suppressed at higher costs. The results show how disturbance source and timing shape exposure and controller burden in this model. They motivate testing adaptive agents with distinct disturbance sources and assessing regulatory activity alongside exposure.

Abstract: Given a rational matrix $A$, is every sufficiently large power $A^n$ entrywise nonnegative? We prove that this problem is decidable in deterministic polynomial time. This is in contrast with deciding eventual nonnegativity of a single prescribed entry sequence $(A^n)_{ij}$, which amounts to the Ultimate Positivity Problem for linear recurrence sequences, whose decidability is open. The previous decidability procedure (D'Costa, Ouaknine and Worrell, STACS 2024) splits the matrix powers into residue classes modulo a torsion exponent $D$ whose value can be exponential in the input size. We show that it is sufficient to check a single progression $A^{Dk+1}$: the exponents $n$ with $A^n \geq 0$ are closed under addition, and two consecutive exponents of the progression are coprime, so they generate every sufficiently large exponent. Along the progression, eigenvalues are grouped by their common $D$th power, and the required coefficients for each group are computed without forming $A^D$, any $λ^D$, or a splitting field: each summand is evaluated in its own root field $\mathbb{Q}(λ)$, and Galois equivariance keeps the degree and height of every class sum polynomially bounded, enabling certified zero and sign tests. The same algorithm with one case rejected decides eventual positivity, and a variant decides whether the matrix has any nonnegative power at all.

Abstract: Interval Markov decision processes (IMDPs) provide a natural framework for modeling stochastic systems with uncertain transition probabilities, represented by probability intervals and resolved adversarially. Such uncertainty arises naturally, for example, when the transition model is learned from finite data or obtained through model-based reinforcement learning. In this paper, we study the automata-theoretic verification of IMDPs against rich temporal specifications, including all LTL specifications, by considering the broader class of ω-regular objectives. We show that classical automata-theoretic verification techniques extend to IMDPs, but with a sharp distinction determined by the structure of the transition intervals. For stable IMDPs, where either the upper bound is zero or the lower bound is strictly positive, verification reduces to ordinary MDP analysis and can be carried out using the standard automata used in that setting (good-for-MDP automata). For unstable IMDPs, where intervals may include zero while the upper bound is strictly positive, verification becomes game-like and requires automata whose nondeterminism can be resolved on the fly (good-for-games automata). Building on these insights, we develop algorithms for verifying ω-regular specifications over IMDPs and derive probabilistic guarantees when the interval model is learned from sampled data. The resulting framework enables principled verification of stochastic systems under probabilistic model uncertainty, connecting automata-based verification with data-driven stochastic modeling.

Abstract: This paper proposes a structured approach to learning linear operators for control systems from data. We address both structural and learning-theoretic aspects of the problem. To derive structural assumptions, we propose using the well-established framework of (semi)groups for evolution equations, as operators in control systems are of the same type. Further, we propose analyzing learning algorithms through the lens of the inverse problems framework. This reveals how a learned model depends on the data via error decompositions, convergence guarantees, and optimal regularization -- enabling us to compare existing methods and derive provably advantageous algorithms. In order to obtain these results, we restrict our scope to bounded operators on Hilbert spaces. Although this may appear restrictive, existing approaches often make this assumption implicitly to obtain matrix-like representations. We demonstrate the power of using these frameworks by deriving a convergent estimator for time-varying systems.

Abstract: Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.

Abstract: We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP

Abstract: We consider the problems of formal verification and synthesis in parametric Markov decision processes (MDPs) with general measurable state and action spaces. The heart of our approach is a parameter flattening transformation, which allows us to transform parametric MDPs into semantically equivalent non-parametric MDPs. Building on this transformation, we introduce the novel notion of parametric supermartingale certificates, which generalize the traditional supermartingale certificates---used for non-parametric MDPs---to the parametric setting. We use our parametric supermartingale certificates to design algorithms for verification and approximate synthesis in polynomial arithmetic parametric MDPs. This leads to the first verification and synthesis algorithms for parametric MDPs with general state and action spaces. We implement our algorithms and experimentally evaluate them on several continuous parametric random walk benchmarks.

Abstract: In recent years, weak-form methods have made significant advances in data-driven discovery of dynamical systems. However, in high-dimensional settings, current techniques can prove expensive in both computation and memory. In this work, we introduce TT-WSINDy, which combines techniques of the Multidimensional Approximation of Nonlinear Dynamics (MANDy) and Weak Sparse Identification of Nonlinear Dynamics (WSINDy) methods, implementing requisite computations in the tensor-train (TT) format. We demonstrate that this method is able to search an exponentially-growing space of candidate functions -- performing weak-form transformation, regression, and sparsification -- without suffering from the curse of dimensionality.

Abstract: The same underlying computational problem is solved across unrelated fields under different names: recursive Bayesian state estimation appears as a "Kalman filter" in control, "Bayesian forecasting" in pharmacokinetics, and "data assimilation" in geoscience. Topical and citation-based scientific embeddings cannot see this shared problem. We distill each paper once into a domain- and method-name-stripped faceted computational fingerprint, a free-text mechanism skeleton plus controlled computational facets. We define a tunable, facet-selectable distance over it. The goal is solution import: surface cross-field pairs solving the same problem, so a bespoke implementation can be swapped for another field's standard, specialized solver. On a benchmark of 18 method families across 109 papers, the skeleton lifts cross-domain retrieval average precision over the abstract from 0.222 to 0.513, and the whole fingerprint reaches 0.557. Strikingly, four trained scientific embedders all fall below plain abstract+TF-IDF: they encode topical and citation similarity, the wrong signal for this task. The gain is the representation: the abstract-to-skeleton swap lifts every embedder, and the pipeline is one cached LLM call per paper plus a cheap embedder. An interventional re-skin / math-edit test shows the fingerprint tracks the computation, not the field. On a 501-paper wild corpus, known twins dominate the top of the ranking (23 of the top 30); with planted pairs excluded from the results, three blind LLM judges rate 3 of the top 5 and 8 of the top 30 pairs genuine import candidates, and 0 of 30 random ones. The human verification is the four executed imports: in one, an open standard solver reproduces a bespoke clinical dosing engine's output. We release the benchmark, the code, and the distillation prompt.