Papers
Machine learning for physics, as it is published.
New work from arXiv on CFD surrogates, neural operators and physics-informed learning, refreshed every hour. Search it, filter by topic, or start with the papers we keep coming back to.
Papers
New work from arXiv on CFD surrogates, neural operators and physics-informed learning, refreshed every hour. Search it, filter by topic, or start with the papers we keep coming back to.
2,365 papers in Neural operators · newest first
cs.CE
This paper proposes a label-free physics-informed neural network (PINN) for non-associated Mohr-Coulomb plasticity with incremental strength reduction for the reliability of spatially variable - slopes. The PINN is trained with a virtual-work loss on the finite-element mesh and solves many random fields at once on a graphics processing unit. With a solver budget fixed on the homogeneous slopes, it reproduces the Abaqus factor of safety within 2 to 3% root-mean-square on 52 random fields and locates the failure mechanism in the same weak zones, with the plastic strain in the band under-resolved. A neural operator trained only on the PINN solutions predicts the factor of safety, displacement path and failure mode of a million fields in seconds. No finite-element solution enters the pipeline: Monte Carlo fields, importance samples and operator training data come from the PINN, and Abaqus serves only to verify the result. Monte Carlo simulation, the first-order reliability method, importance sampling and the operator give consistent failure probabilities down to for the slope studied, but a 3% scatter in the factor of safety inflates a probability of by up to a factor of two, and the absolute values refer to one mesh. A heterogeneous Bishop analysis gives failure probabilities 2.6 and 8.3 times lower although its critical circles follow the PINN mechanism, because a small difference in the factor of safety is amplified in the tail. The operator also yields the failure mechanism and displacement response conditional on failure.
cs.LG
Operator learning methods such as DeepONets and FNOs often struggle with PDE families featuring sharp interfaces, heterogeneous coefficients, and localized multiscale structures. We introduce a partition-of-unity (POU) mixture-of-experts framework for localized operator learning, in which geometry-aware gating networks produce smooth spatial partitions which blend the contributions of local expert networks. Our main contribution is HiRefPOU, a residual-style hierarchical POU architecture for DeepONets that organizes localized representations through nested parent-child partitions while preserving global continuity. We also show that the same POU principle can be incorporated into Fourier Neural Operators to introduce spatial adaptivity without modifying the underlying spectral layers. On heterogeneous Darcy and reaction-diffusion benchmarks, HiRefPOU achieves substantially lower error than global DeepONet and static POU-MoE baselines, while the broader operator-learning experiments show that the benefits of localization depend on the PDE structure and the chosen neural-operator backbone. The learned partitions are interpretable and align with interfaces and regions of rapid solution variation. These results show that explicit geometric localization can improve both accuracy and interpretability in neural operator learning.
cs.LG
Neural operators offer an efficient alternative to classical PDE solvers, but most learn a monolithic map per equation family and discretization. Real systems are compositional: transport, diffusion, wave, and reaction processes can act simultaneously. Existing mixture-of-experts operators typically use sparse top- routing, although concurrent physics is naturally a blend rather than a discrete choice. We propose COSMOS (Cooperative Operator Specialists with Mechanism-level Operator Soft-routing), a soft mechanism-mixture neural operator. Four process-biased specialists remain active at every step and are continuously mixed by a learned gate, with their features fused by a small network. Specialists share a coarse latent grid, while a zero-initialized full-resolution residual restores detail lost through the bottleneck. We also introduce an operator-splitting compositional benchmark with known mixture weights per trajectory. Against family-tuned FNO under 20-step rollouts, an initial 3-seed evaluation suggested gains on diffusion–reaction, parity on Navier–Stokes, and weaker shallow-water performance. An 11-seed audit showed that the diffusion–reaction gain was unstable, motivating caution in small-seed rollout comparisons and precluding a reliable accuracy-win claim on these families. Ablations show that uniform routing or removing the specialist mixture substantially degrades stable-regime rollouts. On the labeled benchmark, dense soft routing yields lower error than hard top-1 routing at identical fusion; using generator weights at inference further lowers rollout error to , diagnosing limitations of the learned gate. However, routing labels do not align with the specialists' intended mechanisms: COSMOS supports compositional accuracy, not mechanism identity.
cs.LG
Fourier Neural Operators (FNOs) offer an efficient paradigm for solving partial differential equations (PDEs). However, FNOs rely on a fixed Fourier basis and hard frequency truncation, which inherently limit their ability to model non-periodic, localized, and fine-scale solution structures. We propose the Stochastic Adaptive Fourier Decomposition Neural Operator (SAFDNO), a spectral neural operator that replaces predefined Fourier modes with an adaptive Takenaka-Malmquist (TM) orthonormal system derived from the theory of Stochastic Adaptive Fourier Decomposition (SAFD). Instead of performing an expensive greedy pole search in classical SAFD, SAFDNO amortizes stochastic pole selection through a neural pole predictor and constructs the adaptive TM system directly from latent features. The resulting operator performs learned filtering on coefficients of analytic branches in the Hardy space with respect to the input-adaptive TM system, preserving the global receptive field of spectral operators while providing a more flexible representation than using fixed spectral bases. Across nine PDE benchmark problems, SAFDNO achieves the best performance on all six regular-grid problems among strong neural operator baselines, with especially notable gains on Darcy, Burgers, and Navier-Stokes, where fixed Fourier modes are often less effective at modeling localized oscillations, sharp transitions, and multiscale structures. SAFDNO also exhibits stronger zero-shot super-resolution performance and shows less performance degradation when deployed on finer discretizations. These results suggest that the input-adaptive TM system provides a promising alternative to fixed spectral representations for neural operator learning.
cs.LG
Graph Neural Operators (GNOs) provide flexible surrogate models for learning solution operators of partial differential equations (PDEs). However, standard GNOs typically parameterize the integral kernel with a monolithic neural network and evaluate kernel interactions over graph edges, leading to substantial computational and memory overhead at high resolutions or with large neighborhoods. To address these limitations, we propose Functional Tensor Decomposition Graph Neural Operator (FTD-GNO), a memory-efficient GNO framework that decouples the high-dimensional continuous integral kernel into low-dimensional mode-wise functions. By instantiating the kernel with classical tensor decomposition formats, including CP, Tensor-Train, and Tucker decompositions, FTD-GNO enables algebraic reconstruction of the integral operator without explicitly materializing full edge-wise kernel tensors. This factorized formulation reduces the memory footprint of kernel evaluation and aggregation while retaining the continuous operator-learning structure of GNOs. Theoretical complexity analysis shows that FTD-GNO substantially lowers parameter and activation-memory costs associated with high-dimensional kernel construction. Experiments show lower peak memory than the corresponding unfactorized graph-integral baselines, with shorter recorded training times. Fourier-graph experiments further demonstrate that FTD can improve the efficiency of a graph-integral layer within a hybrid operator and has good scalability.
cs.LG
High-fidelity finite-element simulations provide accurate crashworthiness predictions, but their cost limits iterative design exploration. Deep learning surrogates can reduce this cost, but many component-level models are developed under a single prescribed boundary condition, limiting generalisation to boundary variations. This work proposes a Boundary-Condition-Aware Transformer Neural Operator (BAT-NO) for autoregressive prediction of transient displacement fields and scalar crashworthiness responses under variations in geometry and boundary conditions. A B-pillar simulation framework evaluates generalisation across variations in geometry, impact position and velocity, and support stiffness. BAT-NO combines recurrent mesh processing with latent-grid Fourier operator processing. Boundary-condition information is transferred to the latent grid through a hybrid local–global mechanism. Slice-based attention models interactions among physically related regions, while direct boundary-to-grid projection preserves local spatial structure. Across the validation sets for the shape-only, shape-and-loading, and shape-loading-boundary cases, BAT-NO achieves the lowest mean final-step mean nodal Euclidean displacement error among the evaluated baselines. In the most challenging case, it reduces the mean error by 32.6% relative to the second-best model. Hyperparameter tuning reduces the validation error from 0.451 to 0.269 mm, with a comparable error of 0.267 mm on 300 unseen test simulations sampled within the investigated design space. An attention-based scalar decoder jointly predicts six response trajectories with a mean relative error of 2.46%. Most derived crashworthiness indicators have median errors below 3%. These results show that explicit local and global boundary-condition representations improve crashworthiness prediction over expanded component-level design spaces.
math.NA
We extend to the Willmore flow the phase-field neural operator approach proposed by Bretin, Denis, Masnou, and Terii (2022) for mean curvature motion in two and three dimensions. The resulting lightweight, single-time-step architecture is motivated by a Bence-Merriman-Osher-type expansion of the Allen-Cahn semigroup, and is trained using trajectories generated by a hybrid minimizing movement scheme. In our experiments with non-oriented interfaces, unconstrained convolution kernels produce anisotropic dynamics, whereas radial kernels enhance the evolution process. We also construct three-dimensional kernels from trained two-dimensional kernels via their radial Fourier profiles, which provides an effective initialization for the non-oriented 3D model. Finally, we show applications to curve and surface reconstruction from unoriented point clouds, including reconstruction using the transferred model without the need for additional 3D training.
cs.LG
Large-scale pre-training has improved the generalization of neural operators across diverse PDEs. However, existing PDE foundation models still struggle with heterogeneous dynamics, where shared representations may cause knowledge interference, while mixture-of-experts (MoE) architectures suffer from increasing expert redundancy. We propose SPEAR, a spectral-disentangled MoE neural operator with knowledge-guided expert aggregation for large-scale PDE pre-training. SPEAR decouples latent features into low- and high-frequency components, enabling shared modeling of transferable dynamics and specialized learning of PDE-specific patterns. To address expert redundancy, we design a knowledge-guided expert aggregation strategy that measures expert similarity from dataset-specific learned knowledge and routing preferences, enabling the identification and consolidation of similar experts. Experiments on twelve PDE datasets and multiple downstream benchmarks demonstrate superior performance in pre-training, fine-tuning, and transfer learning. Furthermore, our aggregation strategy reduces the number of experts by 50% while maintaining or improving prediction accuracy, achieving a balance between model efficiency and generalization for PDE foundation models.
cs.CE
Magneto-active elastomers, as a class of smart materials, exhibit strongly coupled magnetic and mechanical behavior at finite strains. Considering variations in microstructure, material properties, and geometry can lead to computationally expensive analyses. Building on the finite operator learning (FOL) framework, this study develops a data-free, physics-informed operator-learning framework for families of coupled finite-strain magneto-mechanical boundary-value problems. The governing magnetostatic and mechanical equations are enforced through finite-element weak-form residuals, allowing the neural operators to be trained without labeled finite-element solution data. The main contribution of this study is the development of a weak-form staggered neural operator (WSNO) framework for strongly coupled magneto-mechanical saddle-point problems. Separate magnetic and mechanical neural operators are trained alternately using a staggered optimization strategy, while their physical coupling is retained through the constitutive relations and residual evaluations. The resulting framework learns mappings from parameterized material and geometric descriptions to the corresponding coupled magnetic and mechanical solution fields. The proposed framework is investigated across several settings, including heterogeneous random-inclusion microstructures, varying magnetic phase contrast, area-fraction-dependent geometries, strongly out-of-distribution material morphologies, and three-dimensional geometry-parametric problems. In addition, the learned operator is combined with a neural-initialized Newton strategy, in which the nonlinear finite-element solver is initialized using the neural prediction. The results demonstrate that the proposed operator-learning framework can accurately capture coupled magneto-mechanical responses across a broad range of parametric problem settings.
cs.LG
Accurate thermal analysis of heterogeneous 2.5D/3D-IC packages is essential yet computationally prohibitive. A single full-package FEM simulation can take hours, while AI-based surrogates treat the entire stack as a monolithic prediction target and must be retrained whenever the die count or topology changes. To address this limitation, this work proposes Domain-Decomposed AI-Accelerated Iterative Solver for Thermal Analysis (DAIST), a composable thermal solver that decomposes the global package simulation into block-level subdomain problems, replaces subdomain solvers with neural operators, and couples them through iterative exchanges of interfacial temperature and heat flux. This local-to-global architecture eliminates the topology lock-in of monolithic models: block-level neural operators can be directly reused in unseen package assemblies without retraining. The iterative coupling strategy further provides a controllable accuracy-runtime tradeoff, where the iteration budget can be adjusted to trade accuracy for runtime. Evaluated on a multi-chiplet system and an advanced packaging system, DAIST achieves up to speedup over traditional FEM solvers with mean temperature errors of 0.068% and 0.323%, respectively, while demonstrating cross-topology reuse of block-level models across structurally distinct package assemblies.
eess.SY
Neural operators approximate PDE solutions within a geometry family, but independently learned local operators need not form a consistent global simulator. We study frozen, single-pass composition for steady incompressible flow in idealized two-dimensional airway trees. Separate Tube, bifurcation, and trifurcation DeepONets are trained on 4,872 primitive CFD cases using field supervision and auxiliary divergence, port-flux, component-balance, and port-pressure penalties. The validation-selected deployment is frozen before whole-tree CFD fields are inspected and assembled without tree training, iterative coupling, flux correction, or CFD-informed adjustment. It retains major flow patterns and controlled pathology responses with 0.204-0.215 s CPU inference, but has a 22.68% prescribed-inlet-normalized external residual. A post-hoc sensitivity protocol, frozen before new training and evaluation, repeats Data, Div, and Full models across three seeds with Tube fixed. Relative to Data, Full reduces primitive composite scores by 26.7% for Y2 and 26.4% for Y3 and reduces assembled component-residual and interface-mismatch RMS by 7.2% and 16.6%, respectively. Nevertheless, mean tree velocity error increases by 7.5%, while external residual increases from 17.49 +/- 4.38% to 26.77 +/- 3.65%. Local regularization can therefore improve primitive and assembled local diagnostics without ensuring accurate global fields or conservation.
quant-ph
General operator learning for parametric partial differential equations (PDEs) is a fundamental challenge at the intersection of artificial intelligence and physics-based modeling. Physics-informed Deep Operator Networks (PI-DeepONets) incorporate governing equations into learning, but face optimization difficulties in high dimensions. Quantum extensions offer additional representational capacity, yet their trainability can be hindered by barren plateaus. Here, we introduce QPI-DeepONet-MAC, a hybrid quantum-classical architecture that integrates parameterized quantum circuits into PI-DeepONets through a multiplicative-and-additive coupling (MAC) of classical representations and quantum expectation values. We prove analytically that the proposed architecture retains universal operator approximation capabilities. Furthermore, we derive bounds on quantum parameter gradients, establishing explicit conditions to avoid barren plateaus alongside a scaling criterion for trainability. Additionally, we derive a lower bound on the number of sensors required for a target accuracy, directly linking sampling requirements to input function regularity, domain dimension, and PDE dynamical growth. Finally, we propose an informational regularization scheme based on quantum coherence and state fidelity dynamics to prevent coherence loss and preserve quantum structure during training. Together, these results establish a theoretical foundation for quantum-enhanced physics-informed operator learning and position QPI-DeepONet-MAC as a scalable, stable framework for solving parametric PDEs.
cs.LG
Spatiotemporal forecasting on hydrologic graphs is especially prone to instability in open-boundary systems, where the forecast domain exchanges fluxes with an unobserved exterior. In such systems, boundary nodes receive external forcing, e.g., upstream inflows in rivers or tidal signals in coastal regions, that is typically unavailable at prediction time. The absence of this information can compound errors as forecasts unfold in an autoregressive fashion, leading to inferior long-horizon performance. This paper dissects this instability issue by exploring two questions. 1) What boundary forcing enters the forecast domain when information beyond the boundary is missing? 2) How should this forcing propagate through the domain without incurring error amplification under autoregressive rollout? To address both, we propose a new computing framework comprising two key components. First, to compensate for the boundary forcing, our framework learns ghost node proxies from the boundary and interior nodes, striving to approximate unobserved external inputs. Second, to control error accumulation from these learned proxies, we leverage two physics refiners. In particular, one refiner enforces local consistency by aligning ghost proxies with their two-hop neighbors (i.e., boundary nodes and their immediate interiors). The other refiner enhances global stability by correcting the model forecasts through a physics-guided graph neural operator, reducing long-horizon numerical drift. Two real-world hydrologic graphs are employed for empirical evaluation. Comparative results show that our proposal enjoys higher prediction accuracy and long-horizon stability over both learning-based and physics-informed model competitors.
cs.LG
Many scientific and machine learning systems, from molecular dynamics to diffusion models and beyond, are governed by stochastic dynamics with low-dimensional structure, evolving on slow timescales. However, target trajectories, used to identify and interpret such dynamics, are often inaccessible: only biased or static samples that explore the underlying manifold are available. We introduce Langevin-Informed Transfer Learning (LITL), a framework for recovering target Langevin dynamics from biased source samples using only black-box feedback. LITL learns the leading spectral structure of the target infinitesimal generator and the projected drift through Dirichlet representation learning, enabling kinetic reconstruction in spectral form and slow-manifold gradient field estimation. We further introduce a spherical variant well suited to steering normalized latent representations commonly used in learning systems toward desired objectives. We establish finite-sample guarantees for eigenvalue, eigenfunction, and projected drift estimation in Sobolev norms, thereby ensuring generalization of these quantities and their first-order derivatives. Empirically, LITL recovers physical transition timescales from biased molecular simulations, builds kinetic structure from static samples of generative models, reconstructs spherical symmetries of physical systems, and enables post-hoc latent steering of trained neural networks under black-box feedback. Together, these results position spectral operator learning as a practical framework for recovering stochastic dynamics under distribution shift and unlock applications across machine learning and the physical sciences.
physics.geo-ph
We introduce GeoFWI3D, a large-scale open-source benchmark dataset of geologically plausible 3D subsurface models designed to accelerate deep learning (DL) assisted seismic imaging and full waveform inversion (FWI). FWI is a physics-driven, wave-equation-based optimization technique used in seismic imaging to estimate the subsurface properties. However, traditional FWI is highly non-linear, non-unique, and ill-posed. The iterative process often converges to local minima when the starting model is insufficiently accurate. Most importantly, 3D FWI is prohibitively expensive. With the advent of DL, a direct mapping between the shot gathers and subsurface properties can be established by training a network on realistic models. The primary bottleneck for such approaches is the lack of large-scale, realistic training datasets. GeoFWI3D addresses this gap with 10,000 geologically diverse velocity models at 96x96x96 resolution, spanning four structural complexity classes: pure stratigraphy, faulted networks, salt diapirism, and complex coupled fault-salt systems. Each model is accompanied by co-registered multi-modal labels including compressional velocity (Vp), zero-offset seismic reflectivity, relative geologic time (RGT), and semantic fault/salt masks. To facilitate systematic evaluation, we introduce benchmark tasks covering fault detection, joint salt body segmentation and chronostratigraphy prediction, FWI, wavefield and traveltime surrogates using neural operators, and generative modeling with a 3D diffusion model. We present baseline results for each task to establish reference performance metrics for future users. The dataset is publicly available under Creative Commons Attribution 4.0 International.
cs.LG
Operator learning on probability measures can be accomplished with transformers. For measures with polynomial tails, the exponential weighting in softmax can make the corresponding measure-level attention integrals diverge. This motivates replacing the exponential with slower-growing functions. We construct two benchmarks for operator learning on measures with closed-form targets. We use these benchmarks to study attention kernel growth and data transformation in post-norm transformers. Without data transformation, the softmax models exhibit ensemble collapse on both heavy-tailed benchmarks, while the three slower-growing kernels avoid collapse. Symlog preprocessing allows softmax to avoid collapse on the matrix inverse task but not on the sheared swap task. On the Gaussian control, all four kernels perform similarly. We also examine how sample size affects the sensitivity of empirical energy and Wasserstein distances to tail differences. These results support slower-growing attention kernels as an effective design choice for post-norm transformers learning from heavy-tailed ensembles.
stat.ML
Neural operators learn maps between function spaces, while hereditary network dynamics are described by Volterra resolvents with non-rational Laplace symbols. We introduce a fractional Laplace neural operator (fLNO) that embeds this structure in the learned map. For commuting excitation–Laplacian pairs, one block graph-spectral layer represents the full linear Volterra solution operator exactly. We establish an expressivity frontier for finite rational realizations: they approximate fractional memory geometrically on compact frequency windows, but cannot reproduce the non-integer critical asymptotics generated by a branch point, and on the half-line the best rational rate is root-exponential. The same theory yields trainable parametrizations that enforce a prescribed stability margin by construction, and a graphon-transfer theorem separates genuine operator consistency from parameter sharing. In a common-data benchmark, positive rational operators can match or exceed fLNO accuracy on finite horizons, whereas in controlled near-critical experiments fLNO recovers the branching coordinate more faithfully with far fewer parameters; unconstrained rational fits can cross the stability boundary, while certified parametrizations cannot. A four-parameter spectral law transfers without retraining from graphs of size 48 to 192 with 0.51–0.62% relative error. Applications to Chilean aftershock sequences and to renewal models for Chile and 21 Italian regions illustrate structured inference with explicit uncertainty. The contribution is an operator-learning architecture in which exact memory structure, physical coordinates and stability guarantees coexist with competitive accuracy.
cs.AI
Particle transport under multiple scattering is central to radiative transfer and plasma physics, yet high-fidelity Monte Carlo (MC) simulations must trace prohibitively many particles. Learning-based surrogates can amortize this cost, but typically train on expensive, well-converged MC solutions. We propose the Particle Transport Neural Operator (PTNO), a neural operator that learns particle transport surrogates directly from noisy, low-cost MC labels. Such labels pose two challenges: (1) high variance, which destabilizes standard supervised learning, and (2) a high dynamic range (HDR) spanning many orders of magnitude. For the first, we learn the solution operator from noisy labels of many configurations, amortizing MC cost and generalizing to unseen configurations. Because MC labels are unbiased, we show that the squared loss on them shares its minimizer with the loss on converged solutions, and our budget-allocation study over training scenes , MC samples per render , and independent renders per scene shows that many noisy scenes beat fewer converged ones. For the second, a nonlinear transform such as the logarithm biases noisy supervision. Instead, PTNO keeps labels in physical space and enforces positivity with a softplus output layer that represents small values effectively. We further train with a pointwise relative loss (PRelL2), the stop-gradient relative loss of HDR denoising and neural rendering, which normalizes each residual by the stop-gradient prediction instead of the noisy label. We demonstrate PTNO on neutron transport in fusion reactors and radiative transfer in participating media. On the two neutronics tasks, PTNO is - faster than converged MC on the same CPU and - cheaper than MC at matched accuracy; on the two radiative-transfer tasks, MC at matched accuracy costs - as much as PTNO.
math-ph
Traditional simulations of parametric partial differential equations (PDEs) rely on repetitive computations for each parameter, which makes high-fidelity design impractical. Neural operators address this issue by learning solution operators, accelerating parameter-space mapping by orders of magnitude. Recent Transformer-based neural operators attempt to capture global dependencies, but often at the cost of quadratic attention complexity. Transolver resolves this problem by projecting physical states into a reduced slice space for attention computation. Although fast, this projection sacrifices fine spatial information. Moreover, by operating in this reduced space with shared weights across attention heads, it may constrain the model's flexibility, thereby limiting its capacity to capture complex phenomena. To address these issues, we propose the Cluster Attention Neural Operator (CANO), which reformulates attention via a novel cross-attention mechanism that dynamically clusters queries while preserving full-resolution keys and values. This avoids slice compression loss and removes weight-sharing limits. At the same time, the model remains fast without losing global interactions. Empirically, CANO achieves state-of-the-art performance across canonical PDE benchmarks, covering fluid and solid dynamics (e.g., Navier-Stokes, Airfoil, Plasticity), irregular unstructured geometries (e.g., Pipe Turbulence, Composites), and long-term temporal rollouts. Across solid deformation and turbulent flow benchmarks, CANO achieves lower errors than baselines and exhibits strong geometric adaptability and temporal consistency.
cs.LG
Machine-learning surrogates accelerate physical simulation, but lower prediction error need not coincide with lower error in physically relevant flow statistics. We examine this question for flow around a NACA4418 airfoil using paired computational-fluid-dynamics simulations and experimental particle-image-velocimetry measurements. A mean-preserving input intervention removes velocity fluctuations from selected regions of observed flow histories. Across four neural operators, removing fluctuations from the most energetic 10% of valid observed cells changes forecasts more than equal-area random removal. Because the masks are not matched for removed fluctuation energy, this contrast measures sensitivity, not independent evidence of physical importance. Separately, a CNO has lower velocity-field error but substantially higher two-component fluctuation-energy error than the reference on both analysis subsets. An output attenuation stress test also demonstrates disagreement between benchmark errors and domain-summed fluctuation energy. These single-benchmark results motivate reporting complementary physical diagnostics alongside aggregate prediction scores; they do not establish counterfactual physical correctness.