computational fluid dynamics developers

Abstract: Radiative heat transfer plays a dominant role in fire simulations, where solving the radiative transfer equation (RTE) is a computational bottleneck. Neural operators can accelerate these computations by learning solution operators from offline simulation data. However, conventional DeepONet surrogates rely on fixed-grid inputs and affine reconstruction, limiting their flexibility across resolutions and their approximation accuracy at a fixed latent dimension. In this paper, we develop a transformer-based DeepONet for multi-resolution RTE surrogate modelling. The proposed approach combines a Perceiver encoder, which maps temperature and absorption coefficient fields at different spatial resolutions into a fixed-size latent representation, with a cross-attention decoder that enables non-affine reconstruction of the radiative intensity. This architecture allows a single model to process inputs at varying resolutions without resampling them onto a common grid. We formulate a computable empirical version of a previously derived theoretical lower bound for affine reconstruction, enabling direct comparison with model errors on the same dataset. Experiments on a two-dimensional pool-fire dataset demonstrate that the proposed approach outperforms autoencoder-based DeepONets in predicting both radiative intensity and incident radiation, with fewer trainable parameters. Power spectral density analysis shows improved preservation of small-scale structures. Both affine baselines remain above the corresponding lower bound at all evaluated ranks, whereas the proposed model achieves a training error below it at one matched latent dimension, demonstrating the potential of non-affine reconstruction to overcome this representational limitation. Experiments on a dataset combining three spatial resolutions confirm the accuracy advantage of the approach at each input resolution.

Abstract: Partial differential equations (PDEs) are indispensable tools for modeling complex phenomena in physics and engineering. Physics-Informed Neural Networks (PINNs) have emerged as a powerful alternative to traditional numerical solvers by directly incorporating physical constraints into neural network loss functions. Despite their advantages, PINN training requires multi-objective optimization, which often leads to significant gradient conflicts among boundary, initial, and residual loss terms. The Conflict-Free Inverse Gradient (ConFIG) method addresses these conflicts; however, integrating ConFIG with adaptive optimizers such as Adam introduces a critical vulnerability: the conflict-free direction generated by ConFIG is further processed by the optimizer, and the resulting weight update may no longer be conflict-free with respect to the task gradients. To address this issue, Cautious ConFIG is proposed as a concise three-line extension to the ConFIG algorithm that empirically restores the conflict-freeness guarantee prior to the weight update. The method is evaluated on three PINN benchmarks: the 1D Viscous Burgers' equation, the 1D Nonlinear Schrödinger equation, and the 2D Kovasznay flow. Experimental results demonstrate consistent performance improvements over the standard ConFIG baseline. Additionally, the stabilizing effect of cautious masking and its reaction to conflict severity, task number, and optimizer lag are investigated using a simple toy system.

Abstract: Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative $L^2$ error on every problem, reducing error by $10.7\%$--$27.5\%$ relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.

Abstract: The performance of stationary iterative methods is memory-bound, due to the low arithmetic intensity. So, we propose a mixed-precision technique to reduce the amount of data movement in matrix-splitting iterative methods. We demonstrate both theoretically and experimentally that this technique will generally converge at a similar rate as an exact-precision iteration, depending on the precision used and the conditioning of the $M$ matrix. Furthermore, we demonstrate that iterative refinement can be used to refine the solution to full double-precision accuracy. Using this approach, we mixed half and double precision to achieve an average speedup over uniform double precision of $1.60\times$ for scalar and block Jacobi and $1.25\times$ for scalar and block Gauss-Seidel. Furthermore, we reduced the precision in the SSOR iteration used by OVERFLOW, a NASA code for compressible fluid dynamics simulations, and achieved a $1.14\times$ speedup to the overall application.

Abstract: Fast surrogates for tokamak boundary-plasma simulation are typically deterministic regressors mapping a global operating point to a flattened vector of cell values. Near the divertor detachment transition the steady state is not reliably single-valued. A point estimate must average over qualitatively different plasma states, and it arrives with no statement of confidence. Moreover, the flattened vector representation discards the geometric structure of the SOLPS-ITER mesh. This work addresses both problems. We unroll the curvilinear mesh into three fixed-size image tensors whose layout preserves cell adjacency and inverts exactly, letting a convolutional network act on the geometry without loss of information. A conditional flow matching model, well suited to highly sensitive systems, is then trained on this representation. The result is an efficient, scalable surrogate that captures multiple plausible outcomes even at sensitive operating points. Along a gas-puff scan, the predictive distribution splits into a hot and a cold mode across an early regime transition. A further check on synthetic data with an injected bifurcation of known size confirms the model recovers both branches rather than their average.

Abstract: Latent neural operators improve the efficiency of operator learning for partial differential equations (PDEs) by performing the main computation on compact latent representations. However, directly compressing the input representation to obtain such compact representations may discard solution-relevant spatial information, especially for PDE solutions with multiscale structures. To address this problem, we propose HiLNO, a hierarchical latent neural operator that constructs a fine-to-coarse-to-fine latent space and further introduces multi-scale supervision (MSS) and anisotropic Gaussian attention. The hierarchy mitigates potential information loss during compression, while MSS aligns intermediate predictions with downsampled target fields, encouraging solution-relevant structures to be captured across multiple spatial scales. Anisotropic Gaussian attention enables feature transfer across the hierarchy, making HiLNO applicable to general geometries. Experiments on representative PDE benchmarks and a large-scale automotive aerodynamics task show that HiLNO achieves competitive predictive accuracy, while reducing the parameter count by an average of 84.4% and FLOPs by an average of 69.2% compared with LinearNO. Additional experiments demonstrate effective generalization to unseen spatial resolutions. Code is available at https://github.com/JcLimath/HiLNO.

Abstract: Objectives in scientific machine learning are often prescribed as a sum of several terms, such as the residual, boundary, initial, and data losses of a physics-informed neural network. In the pullback-corrected scalar auxiliary variable (PB--SAV) method, one scalar tracks the shifted objective while the component gradients build a positive semidefinite curvature correction of rank at most the number of components. We carry that correction into an optimizer with momentum and an adaptive mobility, applying it to the gradient and the stored momentum in a single implicit solve. A mobility that is nonincreasing in the Loewner order yields an exact modified energy law, covering Euclidean and AMSGrad-type choices; the corresponding identity for momentum appended after the solve carries a cross term of indefinite sign. For a fixed mobility we give a necessary and sufficient condition for local stability at a stationary point, depending on the Hessian minus twice the correction, and show that it also gives local geometric convergence for every scalar relaxation sequence. The implicit solve reduces to a dense system whose order is the number of components. In the forward Burgers comparison, four components reduce the mean tail objective by 64.7% and the final solution error by 50.2% relative to one component at the same learning rate and momentum settings.

Abstract: Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs). Automatic differentiation (AD) plays a central role in this paradigm, which is mesh-free and replaces traditional iterative solvers with gradient-based optimization in continuous space. However, the inherent limitations of AD, particularly in handling higher-order derivatives and discontinuous solutions, pose significant challenges for complex problems. This has motivated a growing number of researchers to explore discretization-based constraints as an alternative path. Yet, the respective applicability of these two paradigms remains largely unexplored. In this work, we conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities. Through a rigorous decomposition of approximation, optimization, and truncation errors, we systematically elucidate the fundamental trade-offs and error-governing mechanisms of both paradigms, as well as two representative network architectures: multi-layer perceptron (MLP) and graph neural network (GNN). Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced, with smaller optimization errors compensating for the truncation errors. Moreover, the more complex the nonlinearity and boundary conditions, the greater the advantage of GNN over MLP. These insights offer a robust practical guideline for configuring neural PDE solvers in demanding engineering applications. Our source data and code are available at https://github.com/guoxing0809/neuropde_analysis.