mechanical engineers

Abstract: Generative models support content creation across text, images, videos, and design disciplines, yet their interfaces rely primarily on natural language. While effective for specifying high-level intent, language can be inefficient and imprecise for fine-grained control, iterative refinement, and open-ended exploration. We present PolyGandr, a visual interface for exploring relationships between latent representations and generated outputs, complementing natural-language model steering. PolyGandr formulates generative control as high-dimensional navigation, enabling users to define starting points, steering directions, and step sizes through a unified map-and-mixing-palette interface. Combining global exploration with local refinement supports both broad design-space navigation and fine-grained output manipulation without repeatedly translating visual intent into prompts. We evaluate PolyGandr across three use cases: StyleGAN2 face generation, Stable Diffusion chair design, and rotational kirigami engineering. A formative study indicates that PolyGandr complements existing workflows by supporting real-time editing while simultaneously providing an overview and means to explore the broader design space.

Abstract: This study presents an inverse analysis to determine the stiffness parameters of a bimodulus constitutive model formulated within a granular micromechanics framework. Full-field normal strain measurements are obtained by Digital Image Correlation (DIC) during three-point bending tests and are used as experimental input to a hybrid numerical optimization procedure. The experimental campaign is designed to ensure that the measured strain fields remain within the elastic regime. Since the identification is formulated as a nonlinear regression problem, a physically informed initial parameter set is constructed to guide the optimization toward mechanically meaningful values. An initial estimate of the compressive stiffness is first obtained from the available manufacturer data. The tension-compression stiffness asymmetry is then estimated from the experimentally measured neutral-axis shift, which provides a macroscopic signature of bimodulus material behavior in bending. These estimates define the initial parameter set, which is subsequently refined through a hybrid optimization procedure based on the full-field comparison of the experimental and numerical normal strain distributions. The identified parameter set enables the numerical model to reproduce more closely both the through-depth profile and the spatial distribution of the normal strain component over the selected surface region. These results demonstrate that full-field normal strain measurements provide an effective basis for identifying the stiffness parameters of bimodulus granular micromechanics models. The largest differences between the numerical and experimental strain fields are observed near the load-application region, where localized strain variations are induced by the concentrated load.

Abstract: Generative modeling for Computer-Aided Design (CAD) remains fundamentally challenged by the need to capture the intricate interplay between discrete topological connectivity and precise spatial geometry inherent in Boundary Representations (B-reps). Existing generative approaches largely formulate CAD construction as a sequence modeling problem, but consequently lack an explicit topological inductive bias. In this work, we present GraphCAD, a geometry-aware generative framework that centers modeling on the Edge-Adjacency Graph (EAG), a novel graph-based CAD representation tailored for generative tasks. GraphCAD adopts a hybrid graph diffusion formulation to jointly model the discrete and continuous components of EAG and introduces a geometry-aware denoising mechanism that renders node attributes into spatial point sets. These point sets are encoded by an SE(3)-equivariant backbone and fused into the node representations, thereby grounding the generative process in physical geometry. To support this study, we construct EAGBench, a large-scale dataset of engineering CAD models. Extensive experiments show that GraphCAD can generate complex CAD models with more than 50 faces and achieves state-of-the-art performance in geometric fidelity and distributional alignment compared with sequence-based baselines.

Abstract: While LLM agents can generate visually plausible CAD assemblies, these designs often fail to fulfill their intended physical functions. We study functional CAD design, a setting where agents interpret underspecified requests to construct and iteratively revise mechanisms that satisfy functional requirements in physical simulation. Verifying these requirements is nontrivial: different candidate mechanisms may realize the same function through entirely different components, controls, and operating sequences. To bridge this gap, we propose FunCAD-Harness, a framework that translates abstract requirements into candidate-specific physical tests and uses simulation feedback to guide design revisions. To systematically evaluate this capability, we introduce FunCADBench, a benchmark spanning 46 mechanism families with interdependent and multi-step operations. Experiments across frontier LLM agents show that FunCAD-Harness exposes subtle mechanical failures invisible to visual inspection, substantially improving functional success rates over standard simulation-augmented baselines. Ultimately, our work marks a step toward CAD agents capable of translating human intent into working mechanisms.

Abstract: Recent advances in general-purpose AI models and agentic systems have enabled them to perform some CAD tasks. A systematic assessment is now needed to establish what these systems can accomplish and how well their outputs meet engineering requirements. This paper presents an empirical study of GPT-6 Astra, a SOTA general-purpose AI model, across four basic CAD tasks: model understanding, generation, editing, and evaluation. We design test cases spanning different levels of complexity and engineering requirements. The results show that GPT-6 Astra can support multiple stages of CAD workflows, with successful outcomes in cases involving nontrivial geometric complexity. This suggests that general-purpose AI is beginning to offer practical value in almost all steps in CAD workflow. However, gaps remain between generating visually plausible geometry and producing valid, functional, and editable CAD models. Recurring errors in freeform geometry, parameter dependencies, and mating relationships between parts limit engineering usefulness. Agentic iterative refinement can improve results but does not consistently resolve critical errors; long completion times further constrain practical use.

Abstract: Residual stress remains in an object even in the absence of external forces or thermal pressure, which, in turn, may cause significant plastic deformations. In case the residual stress creates unwanted effects on the material and so is undesirable, an efficient solution is necessary to track and eliminate this stress. Smoothed Particle Hydrodynamics has been extensively used in solid mechanics simulations and the inherent colour-field generation approach is a promising tracker for the residual stress. In this paper, we propose a way to use the colour-field approach for eliminating the residual stress and prevent the undesirable premature failure of solid objects.

Abstract: We present a method for constructing a simple quadrangulation of the sphere from an unordered set of $n$ direction vectors, with application to parametric volumetric models of skeletal joints. We first select a quadrangulation encoding the joint topology by exhaustively enumerating candidate dual graphs, prioritizing convexity and regularity of the resulting quads. We then realize this topology on the sphere without crossings by solving a conflict-constrained assignment problem with lazy cyclic-order separation, and recover the corresponding primal graph. Experiments on direction sets from spherical Fibonacci sampling of size $3$ to $20$ show that the method robustly produces valid, high-quality quadrangulations across a range of input configurations. The resulting primal graph is directly converted into exact rational Bézier shell and column volumes, yielding a compact and precise volumetric representation of the joint. The approach extends naturally to multi-joint assemblies, enabling the construction of truss-based lattice structures within 3D domains.

Abstract: Topology optimization places material so that a structure resists deformation under given loads. Machine-learning methods often report large speed-ups over classical, non-learned optimization, but a slow classical code or a less stiff learned design can exaggerate them. We re-ran released code for eight methods on one workstation and studied two in depth: Optimize Any Topology (OAT), a diffusion model that generates designs, and a graph-based neural optimizer. Speeding up only the classical code's finite-element solver cut these two methods' speed-ups by up to 20-fold. We then timed the classical code until its design was at least as stiff as the learned one; for OAT, we read this point off the completed runs. Published timings imply a 134x speed-up for OAT on a 256 x 256 grid; our matched single-design comparisons gave 1.5x to 3.5x. At matched stiffness, the graph-based optimizer was slower than the classical code in all six two-dimensional problems. Of the six other methods, two kept their advantage, one shrank, one reversed and two were not comparable. In an exploratory audit of published claims, language models judged that 8 of 28 speed comparisons in randomly sampled, openly accessible articles lacked evidence of equal design quality; our own review counted 11, so the count depends on who judges. We provide a comparison procedure and a minimum report: classical code, hardware, design quality, what each timing includes, and data-generation and training costs.

Abstract: CAD segmentation datasets are predominantly synthetic or automatically labelled, leaving it unclear how well existing methods transfer to real-world engineering tasks. Fastener-relevant hole detection represents one such task, requiring human annotation and exhibiting greater geometric complexity than commonly used benchmarks. We address this gap through three main contributions. First, we develop two task-specific rule-based methods, one graph-wire-search and one multi-view based, and complement them with a human-annotated benchmark dataset derived from engineering CAD data that also enables training of UV-Net and AAGNet, representative learning-based approaches for B-Rep segmentation. Second, we conduct an extensive experimental study including parameter optimisation, evaluation on our benchmark dataset, cross-domain generalisation on a synthetic dataset, and an assessment of the feasibility of the approaches at varying dataset sizes and IoU thresholds. Third, we provide evidence-based insights into the strengths and limitations of different approaches: rule-based methods generalise well without training data, with the graph-wire-search method favouring recall and the multi-view method favouring precision, whereas learning-based methods achieve strong performance with sufficient annotations but exhibit reduced robustness under domain shift. These findings highlight trade-offs between data requirements, and robustness, and provide guidance not only for hole detection, but more broadly for real-world CAD segmentation tasks.

Abstract: Large-model-based point-to-CAD generation holds immense potential for advancing industrial design and enhancing 3D modeling efficiency. However, most existing methods approach the problem as a general point-cloud encoding and token prediction task, neglecting the tokenization and supervision specifically for CAD-related primitives. As a result, these methods often struggle to accurately reconstruct the intricate primitive structures. To address this limitation, we propose PrimitiveCAD, a novel multi-stage paradigm for point-to-CAD reconstruction that enhances the geometric accuracy of generated CAD models while better preserving critical geometric features. First, we introduce a primitive-aware point cloud tokenization model, enabling the system to learn more robust geometric representations from CAD point clouds. Next, we perform supervised finetuning on a large language model (LLM) and introduce an operation alignment loss to align key CAD operation frequencies, thereby improving the preservation of global shape features. Finally, we incorporate reinforcement learning (RL) and introduce a feature-line alignment reward to further reduce stochasticity and enhance the fine-grained preservation of geometric features. Experiments on the DeepCAD and Fusion360 datasets show that our method achieves state-of-the-art performance in code validity, geometric accuracy, and geometric feature preservation.

Abstract: Neural operators can accelerate repeated nonlinear mechanics calculations, but their accuracy can deteriorate as operating conditions move beyond the training range. This work studies whether high-fidelity solutions acquired during use can be reused to adapt a neural operator and improve subsequent mechanics-based command selection. Two hard-magnetic soft-material systems are simulated using high-fidelity finite-element (FE) models, providing reference solutions for evaluating surrogate predictions and selected commands. A neural operator predicts deformation from known material, loading, and magnetic-field inputs, while an empirical error estimator determines which predictions may be used for command selection. Selected FE evaluations supplement these predictions, and their complete loading paths are retained for periodic updates of the neural operator and estimator. In both examples, the fixed operator loses substantial accuracy when stiffness and loading move outside the training range. Updates using 16 acquired paths recover much of the lost accuracy while preserving accuracy in the nominal regime. Under the same FE evaluation budget, the updated operators also improve command selection, although the benefit varies with the operating condition. Error estimation is less consistent, with inaccurate predictions sometimes accepted and accurate predictions rejected. These results demonstrate that reusing high-fidelity loading paths can extend the useful operating range of a neural operator. However, improved forward accuracy alone does not guarantee reliable prediction acceptance, highlighting prediction-specific error assessment as a separate requirement for trustworthy decision making.

Abstract: Large Language Models (LLMs) can contribute useful engineering knowledge to robot design, but directly generated designs may rely on implicit assumptions and provide no guarantees of feasibility or optimality. These assumptions are critical because different reasonable modeling choices can materially change which designs are predicted to be feasible or optimal. We therefore present a framework that uses LLMs to construct explicit engineering models containing physical relationships, compatibility constraints, and objectives, allowing these modeling choices to be inspected and revised before formal optimization. The model can then be updated with additional engineering, manufacturer, or system-specific information before formal multi-objective optimization provides feasibility and Pareto-optimality guarantees with respect to the finalized model and specified design space. We evaluate the framework on quadcopter and line-following robot component-selection problems. Across 30 direct LLM design trials, none could be verified as feasible under the corresponding finalized model. Comparisons with an independently developed expert model and successive stages of model refinement further showed that changes in modeling assumptions substantially altered the predicted feasible and Pareto-optimal design sets. Together, these results show that using LLMs to construct explicit engineering models makes the underlying design choices available for inspection and revision before those assumptions determine the optimized designs. Explicit modeling therefore provides an interface for combining LLM-generated engineering knowledge, system-specific information, and formal design optimization.

Abstract: Parameter estimation for ordinary differential equation (ODE) models is a fundamental task that is often complicated by the limitations of conventional optimization-based methods. In theory, differential-algebraic approaches offer an appealing alternative: they reduce the problem to polynomial system solving and do not require user-supplied initial guesses for parameter values. In practice, however, algebraic methods have been limited by their sensitivity to measurement noise, because they require accurate derivatives of observed outputs. In this work, we integrate Gaussian Process Regression (GPR) into the differential-algebraic method and derive a first-order error analysis in terms of noise level and algebraic sensitivity. We evaluate the method across several noise levels on a benchmark of 25 dynamical systems arising in applications including mechanical engineering and systems biology. The proposed method achieves the highest aggregate performance among the methods considered, recovering all sought parameter values and initial conditions to within 10% relative error in 88.5% of runs. These results demonstrate that robust derivative estimation can make differential-algebraic parameter estimation practical for dense, noisy synthetic data while retaining key advantages of the algebraic formulation.

Abstract: Configuration space~(C-space) of a mechanism is a real variety describing the set of feasible configurations that it can attain. To understand the behavior of a mechanism, it is crucial to identify and scrutinize especially the singular points of its C-space. They usually appear when the variety intersects itself, leading to different branches of motion. There exist many approaches to detect those intersections if they are transversal. However, the problem remains challenging if there are tangential, cuspidal, inter-dimensional or a combination of these intersections. This paper exploits an approach acquired from tropical geometry to analyze the neighborhood of any point on C-spaces of 1-degree-of-freedom~(\emph{dof}) mechanisms. This is done by finding the approximate rational parametrization of the curve(s) passing through the given point using Puiseux series. The proposed approach is shown to succesfully detect the transversal branchings in two foldable four bar mechanisms and a cusp in the configuration curve of the double Watt mechanism.

Abstract: Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.

Abstract: Natural load-bearing and transport networks are not assembled in a single step; they emerge through a temporally ordered process of growth, branching, reinforcement, and loop formation. Inspired by this developmental logic, this work introduces a morphogenetic graph-generation framework for mechanical lattices in which a discrete dot matrix provides potential nodes and the final architecture is created by sequential cross-layer and intra-layer growth. The same rule is visualized in two dimensions as a leaf-vein-like developmental sequence and implemented in three dimensions on a 3x3x3 nodal matrix containing 27 candidate nodes. A dataset of distinct three-dimensional lattices was evaluated by beam-based finite element analysis and represented directly as graphs. A graph convolutional neural network (GCNN) with three graph-convolution layers and dual global pooling learns the topology-property mapping and predicts effective compressive stiffness. Coupling the GCNN surrogate with rapid structural sampling enables inverse design: for a target stiffness of 1000 MPa, the selected design was predicted at 1042.43 MPa and validated by finite element analysis at 1027.49 MPa. Beyond straight members, the framework has also been extended to parameterized horseshoe-shaped curved beams made of nonlinear materials, enabling topology-geometry design toward prescribed deformation shapes. Our work provides a paradigm for augmenting the database of mechanical metamaterials, and the resulting perspective links biological morphogenesis, graph learning, and nonlinear shape programming in a unified generative design framework for architected materials.

Abstract: Finite Element Analysis (FEA) is widely used for transient mechanical simulations, but its high computational cost limits real-time and high-resolution applications. Deep learning surrogate models can reduce this cost; however, many existing approaches are restricted to steady-state prediction or cannot jointly predict Node- and Element-based Outputs (NEO) over time. The state-of-the-art DeepFEA framework has addressed these issues but remains limited to structured finite element (FE) meshes. To overcome this limitation, this study proposes DeepFEAv2, a deep learning surrogate framework that enables prediction of transient FEA simulations across different FE mesh topologies and element types. The main contributions of DeepFEAv2 are: (a) a module that uses the FE connectivity matrix to organize input features by element and arrange them into an input sequence guided by the mesh topology; (b) a novel neural network architecture designed to process the input sequence and jointly predict NEO over time; and (c) a FEA-informed optimization strategy for regularizing these NEO predictions. DeepFEAv2 was evaluated on structured and unstructured 3D linear elastic datasets, as well as on a pressure-driven aortic valve dataset. DeepFEAv2 achieved R^2 values up to 0.99 and normalized errors as low as 0.38%. Compared with DeepFEA, it achieved up to 38.0% relative increase in R^2 and up to 87.1% reduction in normalized error. DeepFEAv2 also performed inference up to three orders of magnitude faster than traditional FEA. These results demonstrate that DeepFEAv2 can efficiently model transient FEA simulations across increasingly complex FE settings, providing a scalable surrogate framework for transient FEA.

Abstract: Generating parametric CAD models requires accurate geometry and stable feature dependencies. Existing methods face challenges in selecting geometric references, interpreting sketch-plane local coordinates, and establishing sketch constraints to projected external geometry. We present Vision2CAD, a visual agent harness that combines vision-language model (VLM) reasoning with deterministic CAD kernel operations. An ID-based interface supports explicit geometry selection, a local-coordinate bridge converts view coordinates into sketch coordinates, and projected-edge localization supports external sketch constraints. These mechanisms establish feature dependencies within the supported modeling operations and constraint types. We also introduce the Geometry Explicit Reference Dataset (GERD), which aligned commands, geometry states and IDs at every modeling step. On GERD-EVL and a DeepCAD test subset, Vision2CAD improves mIoU by 11.1\% and 5.6\% and reduces Chamfer distance by 17.3\% and 41.8\%, respectively. Parameter-editing experiments and ablation studies further proved the preservation of parametric dependencies.

Abstract: Digital Twins (DTs) are widely adopted across a variety of application domains. In industrial sectors, particularly in mechanical engineering, they accelerate product development, reduce risks, enable early issue prediction, and lower sustainment costs . In practice, DTs increasingly integrate physics-based (deductive) and data-driven (inductive) models into hybrid models combining the complementary strengths of both modeling paradigms. In this paper, we refer to this integration paradigm as hybridization. Despite this trend, the engineering of hybrid DTs that is, remains insufficiently documented. Hybridization is often introduced in an ad hoc manner, and its implementation is only partially made explicit, which limits reproducibility and transferability. This paper reports on the development of an existing fluidic loop digital twin at Centre Technique des Industries M{é}canique (CETIM). The DT is described using the characterization framework of Gil et al., providing a structured view across its lifecycle dimensions. To make hybridization explicit, the case is further analyzed through a complementary characterization structured along two dimensions: motivation and realization. The resulting description provides a traceable account of hybridization decisions and supports the documentation and transfer of hybrid DT engineering practices.

Abstract: This paper presents a low-clearance hinge joint mechanism based on the 3D printing on sheet fabrication method. This approach simplifies the fabrication of hinge mechanisms and overcomes limitations of conventional origami manufacturing by eliminating the need for adhesives commonly used during assembly, making it suitable for robots at the tens-of-centimeters scale. The advantages and disadvantages of three types of hinge joint mechanisms are compared, and a hinge joint that can be designed with low clearance for various facet thicknesses is selected. Based on the selected hinge joint, the twisting angle and bending force are analyzed, leading to the implementation of a clearance of 0.1 mm. Torsional resistance is experimentally evaluated to measure the torque required for twisting caused by plastic deformation and clearance. The results show that the torque associated with plastic deformation is sufficient to constrain the undesired degrees of freedom of the hinge joint, while the torque required for twisting due to clearance is minimal. Based on the analyzed data, the proposed hinge joint mechanism is applied to a 3-degree-of-freedom delta robot manipulator, demonstrating precise motion with low clearance.

Abstract: Finite element stress fields often exhibit strong local non-smoothness, where stress concentrations near holes, notches, and loading regions induce sharp spatial gradients and high-frequency graph components. Although graph neural networks naturally operate on irregular finite element meshes, conventional message passing is inherently smoothing and progressively attenuates such high-frequency information. Unitary propagation alleviates this problem by preserving spectral magnitudes, but typically relies on matrix functions and high-order approximations with $O(Ked)$ propagation complexity. We propose LiftGCN, an efficient spectrally stable graph network based on Joukowski spectral lifting. LiftGCN maps the real spectrum of a normalized graph operator onto the unit circle through the Joukowski relation and realizes the resulting spectral transformation as a simple second-order recurrence, avoiding matrix exponentials, eigendecomposition, and high-order polynomial truncation. We show that the linear Joukowski backbone has unit-modulus characteristic roots and admits an energy-preserving structure under a positive-definite metric, preventing exponential attenuation of graph-frequency components with depth. Each layer requires only one sparse neighborhood aggregation, yielding $O(ed)$ propagation complexity, while lightweight local nonlinear residuals provide expressive feature transformations. Experiments on finite element stress prediction demonstrate that LiftGCN achieves competitive overall accuracy while improving reconstruction of stress concentrations and local high-gradient structures with substantially reduced computational cost. Our code is available at https://github.com/ChenZeng001/LiftGCN.

Abstract: Solving inverse problems with differentiable physics simulators holds the potential to revolutionize scientific discovery and engineering design, as it enjoys both the strict physical correctness from rigorous numerical physics simulators, and the high efficiency and effectiveness from automatic differentiation and gradient-based optimization. However, currently, this paradigm faces performance issues in optimization. In this work, we target benchmarking the performance of different optimizers to solve various inverse problems. We construct 12 differentiable physics simulators spanning physics domains including discrete mechanics, continuous mechanics, atomistic simulations, rendering, and semi-empirical physics models. Based on these simulators, we design corresponding inverse problems that can be categorized into parameter identification, inverse design, and optimal control. Finally, we conduct extensive experiments to compare the performance of different optimizers, including regular first-order methods, approximate second-order methods, as well as global optimizers, on these inverse problems, and analyze the results to provide insights on how to choose and design optimizers for differentiable programming. We hope such benchmarks can inspire the development of more effective optimizers, and further promote the applications of differentiable programming in various scientific and engineering domains.

Abstract: Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains $C^0$-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks (PINNs) represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element (FE) Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below $3.58\%$ across a stiffness-contrast sweep spanning $(E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}])$, where a strong-form PINN degrades to $5.58\%$, and its displacement error reaches $7.66\%$ against $0.49\%$ for the PI-GNN. A trained network halves the ($σ_{xx}$) error of an energy-based PINN ($5.01\%$ versus $10.94\%$). Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.

Abstract: This paper develops an arbitrary-order kinematic construction that links serial propagation, parallel-mechanism closure, and rigid-platform point fields within one dual screw framework. A cylindrical joint is retained as one native physical block, with revolute and prismatic joints obtained as special cases. For each fixed joint axis, ordinary Bell polynomials organize the derivatives of the exponential factor; across a chain, the noncommuting factors remain in their physical order. Initial-frame prefix and terminal-resolved covariant formulas then produce equivalent representations of the serial twist jet. For a parallel mechanism, repeated Leibniz differentiation, with joint-level derivatives organized by Bell polynomials, yields an arbitrary-order triangular active-passive closure recurrence: the same passive Jacobian is solved at every derivative order at a regular configuration, while the right-hand side contains only prescribed active data and lower-order jets. The resulting platform twist jet is mapped exactly to the point-independent affine invariants of the velocity, acceleration, jerk, and snap fields. The validation is deliberately complementary: a generic 3C chain with noncoplanar axes and nonzero rotational and translational cylindrical coordinates tests ordered serial propagation, an RR+RRR spherical wrist tests active-passive closure, and a Hunt-type 6-RUS mechanism with six active revolute joints tests an independently reconstructed platform jet and its affine fields. Independent differentiation of the rigid motion, evaluation of the affine fields, and the differentiated branch closures all agree through fourth order with residuals below $10^{-12}$ in the corresponding SI units. The formulation is purely kinematic and applies at configurations where the selected active-passive partition is regular.

Abstract: Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.