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.LG
PDE foundation models enable data-efficient adaptation but are deployed autoregressively, so short-horizon accuracy need not imply reliable simulation. We ask whether the short-horizon advantage of a foundation-model recipe persists when its predictions are repeatedly fed back as inputs for a long horizon. We define error amplification as G=E(H_f)/E(H_s), which measures full-horizon error relative to short-horizon error. We evaluate seven models on seven two-dimensional time-dependent PDE datasets: scratch-trained FNO and DeepONet; fine-tuned DPOT, GPhyT, Poseidon, and Walrus; and frozen in-context VICON. We report VRMSE, relative L_2, and relative MAE. Six datasets provide paired short- and full-horizon scores, while CE-RM contributes short-rollout sweeps only. At least one fine-tuned foundation model outperforms both scratch baselines in 26 of 30 short-rollout dataset–budget cells under every metric. Nevertheless, full-horizon VRMSE exceeds short-horizon VRMSE in 140 of 144 paired trained configurations, with a median error amplification of . The count below VRMSE one falls from 104 to 53, and 12 full rollouts exceed ten although none does at the short horizon. Walrus achieves the lowest full-horizon VRMSE in 16 of 24 cells, yet its median error amplification is . Relative L_2 and relative MAE show the same qualitative pattern. These results show that favorable short-horizon rankings can coexist with substantial error amplification. PDE foundation-model benchmarks should therefore report horizon-resolved absolute errors, amplification statistics, and failure incidence rather than a single local forecast score.
cs.LG
Despite advances in numerical methods, non-Newtonian flow simulations remain computationally expensive because velocity and stress must be advanced together through rate- and history-dependent constitutive dynamics. Neural-operator surrogates reduce inference cost after training, but their training data require the same expensive simulations they aim to replace, making data-efficient learning essential when only a few non-Newtonian simulations can be afforded. Here, we introduce the Rheological Operator Pretrained Transformer (RheOPT), which pretrains a transformer-based neural operator on inexpensive Newtonian simulations and fine-tunes it on a small number of non-Newtonian samples. Rather than transferring the Newtonian constitutive response itself, RheOPT reuses geometry-conditioned transport structure that remains useful after the stress model changes. We evaluate RheOPT in two geometry-matched settings: a contraction channel for Giesekus, Oldroyd-B, and Bingham fluids, and the oscillatory wake behind a triangular obstacle for Oldroyd-B flow. In the contraction, Newtonian pretraining substantially improves few-shot prediction, and a model fine-tuned on four target samples can match or exceed a from-scratch model trained on four times as many. This benefit is reliable under reversible instance normalization (RevIN) but dataset dependent under Global normalization, a difference we trace to the dynamic range of each fluid's velocity and stress fields. In the wake, where relative error alone cannot verify that vortex shedding is recovered, phase-insensitive diagnostics show that pretraining recovers the shedding frequency and wake fluctuation amplitude that are suppressed by the from-scratch model in the few-shot regime. Overall, RheOPT establishes Newtonian simulation as a physically grounded pretraining source for data-efficient non-Newtonian operator learning.
physics.geo-ph
Frequency-domain wavefield modeling underpins seismic imaging and inversion, where repeated Helmholtz solutions become costly as velocity models, source locations, and frequencies vary. Neural operators learn mappings between physical fields, and their accuracy can decline outside the training distribution. Diffusion models learn wavefield distributions and generally require repeated sampling evaluations to produce accurate fields. A conditional mean flows model is introduced for complex scattered-wavefield generation. The analytical background wavefield serves as a condition alongside the velocity model, supplying spatially organized information about the source location and frequency. Learning the scattered wavefield avoids the point-source singularity in the target and supports generalization across these physical conditions. The model estimates the average transport velocity over a finite interval connecting Gaussian noise to the scattered field. This representation captures the accumulated displacement and permits both full-interval one-step generation and interval-partitioned sampling. Trained on OpenFWI, the model is evaluated on in-distribution and out-of-distribution velocity structures, frequency generalization, and inference efficiency. In most evaluated settings, one-step mean flows sampling achieves lower wavefield error than multi-step diffusion. Because each sampling step requires another network evaluation, accurate one-step generation also reduces inference time. These results indicate that mean flows can provide accurate and efficient frequency-domain scattered-wavefield generation.
cs.CE
When the parameter of a PDE surrogate is the flow domain itself, the key/value set of an attention model must encode that domain. We present CARS (Coronary Artery Rapid Surrogate), a learned surrogate for the finite-volume solution map of the steady incompressible Navier-Stokes equations on coronary lumens built from image-derived centerlines. CARS maps a centerline with per-point lumen radius and a scalar inlet flow rate to pressure and velocity at arbitrary query points. The attention focal set is the case's own centerline, so geometry fusion costs rather than in the number of evaluation points. Trained per territory on OpenFOAM solutions with synthetic focal lesions, CARS reaches relative errors of (pressure/velocity) on left-coronary and on right-coronary test sets, in ~ms per case against a solver wall-clock of -~s. It outperforms four parameter-matched neural operators, and accuracy saturates eightfold below the operating focal count. Fractional flow reserve derived from the predicted pressure matches the CFD label at Pearson and reproduces the decision in of held-out cases. The predicted velocity does not satisfy continuity closely enough for derived quantities, and the weights are territory-specific.
math.AP
We design a neural operator for the two-dimensional Boussinesq system with anisotropic dissipation on the periodic box, guided by a rigorous analysis of its long-time behavior. The operator maps a small perturbation of hydrostatic equilibrium, of amplitude , to the vorticity at a late time. We prove that the flow relaxes to a stratified shear flow, with the horizontally averaged vorticity staying within of its initial value and the oscillation about it of order . Accordingly, the network outputs plus these two parts, each scaled to its size, which grows as and , respectively. Without this guidance the training either does not converge or misses the growth. With it, one network reproduces, sample by sample on test runs over a factor of in amplitude, the quantity bounded by the theorem to a median error of percent. Being fast and differentiable with respect to the initial data, the network is also used to search the space of initial data, and with every candidate checked by the solver the search finds initial data for which this quantity is times the largest value among the test runs, raising the lower bound for the constant of the theorem by that factor.
cond-mat.mtrl-sci
Long-range interactions in machine-learning interatomic potentials are commonly treated by coupling learned sources through either a prescribed electrostatic kernel or a learned but configuration-independent one. Whether such a kernel suffices depends on the underlying screened interaction. We address this question by using Fourier neural operators (FNOs) to learn how latent source fields, built from the descriptors of a local backbone (here MACE), interact across the simulation cell. For a synthetic ionic system with tunable screening, a prescribed Coulomb kernel underestimates the long-wavelength charge fluctuations by 43% under strong screening, whereas the FNO reproduces them within 3% and recovers 95-99% of the variance of the residual left by the local backbone. A linear FNO, which learns a configuration-independent kernel, performs equally well in the homogeneous case. Under inhomogeneous screening, rearranging the remote medium modulates the coupling between distant ions, a dependence that no configuration-independent kernel can represent. Among the models tested, only the nonlinear FNO responds to this modulation, and its force errors are 27-41% lower than those of configuration-independent models. In first-principles benchmarks of liquid water, lithium lanthanum zirconate, and Au2/MgO(001), MACE-FNO is comparable to a prescribed-kernel model for the bulk systems and more accurate for the interface. It further captures the reversal of the Au2 wetting preference induced by remote Al dopants, with deviations below 5 meV per cell from density functional theory. These results indicate that a configuration-dependent kernel, as learned by FNOs, becomes necessary once the screening medium varies, even though the energy remains quadratic in the charges for a fixed medium.
cs.LG
We present a local and sparse kernel method for learning the solution operators of incompressible flows. It builds on our earlier property-preserving kernel method, whose global output reconstruction and dense operator regression scale poorly. Our method replaces the global output reconstruction by local divergence-free interpolants with divergence-free polynomials of any degree, solved in parallel on the GPU. These interpolants are divergence free, periodic on periodic domains, and, with a multiscale kernel, approach Kolmogorov statistics under refinement. A compactly supported kernel, positive semidefinite in any input dimension, makes the operator regression sparse. Its support radius, and hence the sparsity, is fitted to data, and optional affine polynomials in the input capture global trends. Operator Lagrange weights join the two. They cost one sparse solve per new input and act on the training fields before the output reconstruction, so their local coefficients are never formed. We show that, unlike direct evaluation, Lagrange evaluation does not amplify solve errors by the cancellation among spatial coefficients. On ten fluid benchmarks, validation selects sparse matrices nearly everywhere, with densities that mostly decrease as the training set grows, and affine augmentation lowers the error by more than an order of magnitude for forced isotropic turbulence. These sparse models are at least as accurate as our earlier dense method and more accurate than neural operators. Lagrange evaluation is as accurate as direct evaluation with local output reconstructions, and up to nearly three orders of magnitude more accurate with global ones. We also present GPU timings and storage.
cs.LG
Planning under uncertain control execution noise requires reliable feedback policies across diverse environments. We introduce Planning Under Noise (PUN), a neural operator that learns noise-aware value fields for planning in unseen geometries. Building on an established screened-Poisson formulation of stochastic optimal control, we address two challenges: specifying boundary conditions that support navigation across varying geometries and efficiently computing fields with reliable gradients. Zero-Dirichlet obstacle boundaries terminate sampled paths at walls, weakening the goal signal along long routes and making it harder to estimate reliably. We instead impose reflecting Neumann conditions on obstacles and a Dirichlet condition at the goal, allowing sampled paths to continue exploring free space until they reach the target. To solve the resulting PDE without a configuration-space mesh, we extend a Monte Carlo (MC) solver, Walk on Stars (WoSt), to higher dimensions. However, local MC fluctuations can reverse estimated gradient directions and cause planning failures, making direct planning require costly sampling. PUN amortizes this computation by learning geometry-to-field mappings from noisy, partially converged Monte Carlo labels, avoiding expensive convergence for every training example. At inference, test-time scaling uses bounded local search guided by the predicted global field. We evaluate PUN on planning tasks across dimensions. Across four noise levels, PUN matches or outperforms all compared planners, achieving 95-100% success on the 4D replanning benchmark while maintaining the lowest runtime. At the highest tested noise level in 4D, PUN achieves 10.7-60.6% relative gains in success rate over other methods while running 3.29-32.15 times faster.
cs.LG
Solving dynamical systems in complex and diverse geometries remains challenging for both numerical solvers and learning-based surrogate models. We propose Neural Operator Splitting, a framework for learning fluid dynamics while preserving the physical structure of fractional-step numerical schemes. Instead of learning the full-step evolution monolithically, we decompose each time step into physically interpretable steps, and learn each operator separately using intermediate states from the underlying numerical solver. Guided by the linearity properties of the fractional steps, we introduce a conditional-linear Fourier neural operator (CL-FNO) whose linear state mappings are modulated by nonlinear conditioning on inputs such as boundary geometry. We further introduce two complementary strategies, symmetry sampling and corrective cycling, to mitigate error accumulation during autoregressive rollout. Experiments on 2D and 3D fluid flows with varying geometries and initial conditions show that our full model reduces mean relative rollout error by 68% in 2D and 40% in 3D compared with a single full-step neural operator. At comparable error levels, neural-operator inference is approximately times faster per step than the Monte Carlo solver.
cs.RO
We develop an uncertainty-aware, operator-learning-based surrogate for tendon-driven continuum robots (TDCRs) that generalizes across robot design and actuation while predicting the complete equilibrium configuration. Our central idea is a nonstationary Gaussian-process (GP) kernel whose metric is learned by a shallow neural network from mechanics-aware design coordinates. The network learns how similarity changes across design space, while the resulting kernel defines the design-to-pose operator, its posterior uncertainty, and a criterion for sequential simulator acquisition. We represent backbone position and rotational strain with spatial kernels and reconstruct orientation directly on SO(3), preserving valid material frames and tendon geometry. We use ASkotch with Nystrom preconditioning for scalable kernel operator solves and a separate Nystrom approximation for posterior covariance. Experiments show that kernel-coverage acquisition reduces mean trajectory error by approximately 15% relative to IID sampling, while posterior variance increases for increasingly unsupported designs. Compared with prior neural-operator TDCR models, the method is 30-50% more accurate using approximately 5x fewer training designs while additionally providing posterior uncertainty estimates.
cs.ET
Neuro-symbolic artificial intelligence (NeSy AI) combines neural perception with symbolic reasoning, enabling logical reasoning. Hyperdimensional computing (HDC) is promising for NeSy AI, owing to its high parallelism and strong robustness to noise. However, the dense neural operations and memory-intensive symbolic inference in NeSy impose a severe "memory wall" bottleneck. Existing NeSy accelerators either focus solely on symbolic kernels or rely on conventional digital architectures, lacking a unified and energy-efficient hardware substrate. Moreover, exploring optimal hardware design configurations often requires costly trial-and-error. In this work, we propose UnO-NeSy, a unified CiM framework for optimizing NeSy computing. UnO-NeSy consists of UnO-CiM, an ultra-low-power NeSy architecture based on a homogeneous 2FeFET–2R compute-in-memory (CiM) array that supports both neural computation and HDC reasoning, and UnO-Sim, a cross-workload, end-to-end CiM simulator that models device non-idealities, neural quantization effects, and HDC execution behavior for accurate system-level evaluation. Built on UnO-Sim, we further develop an automated optimizer to identify workload-specific operating points under coupled energy, accuracy, and hardware constraints. By optimizing the operating configuration of the baseline 2FeFET–2R design, UnO-NeSy achieves higher energy efficiency and lower full-system inference energy. Across seven RAVEN configurations, UnO-NeSy achieves average reasoning accuracy under hardware non-idealities, demonstrating a practical and scalable path toward ultra-low-power NeSy hardware.
cs.LG
Neural operators offer an efficient data-driven approach to predicting complex automotive aerodynamic flows. However, spectral bias limits their ability to reconstruct fine-scale flow variations around geometric features such as side mirrors and wheel arches. Reliable assessment of local geometric changes therefore requires models that capture fine spatial detail together with global flow interactions. Inspired by iterative refinement and state evolution, we propose CIS-Transolver, a cross-layer incremental-state neural operator that integrates local multiscale information into global flow modeling. The method extracts local feature variations and encodes them as zeroth- to second-order geometric moments. These encodings are combined with information from earlier layers to form a cross-layer memory. Memory-derived corrections enter the value branch of Physics-Attention, allowing local details to inform global physical-state interactions throughout the network. We systematically evaluate CIS-Transolver on three public automotive aerodynamics benchmarks, ShapeNet-Car, AhmedML, and DrivAerNet++, covering surface pressure fields and volumetric pressure and velocity fields. CIS-Transolver achieves the lowest mean relative L2 error among the evaluated methods for all five prediction metrics across the three benchmarks. Relative error reductions average 8.72% compared with the best baseline for each metric. Also achieves the lowest high-frequency spectral errors across all five metrics, with an average relative reduction of 16.06% compared with the best baseline. These results support cross-layer reuse of local information for fine-scale automotive flow prediction.
cs.LG
Online robust zero-sum Markov games pose a coupled operator-learning and game-solving problem: from nominal interaction data, the learner must recover a worst-case minimax Bellman operator while maintaining strategic coverage and controlling stage-game approximation error. Existing robust reinforcement-learning theory does not yet provide such an operator-learning perspective beyond specialized multi-agent settings. We develop an optimistic dual robust Bellman framework that combines dual robust Bellman fitting, optimistic minimax planning, backward-validated confidence sets, and exploiter-mix data collection. Our analysis is organized around the robust Bellman residuals induced by the candidate value class and yields a regret reduction that separates exploration complexity, robust operator-estimation error, stage-game solution error, and misspecification. Under explicit interpretable assumptions, we obtain conditional theorem interfaces for general and projected function approximation, together with closed results in several structured regimes, including tabular, epochwise cross-fit linear, self-normalized pure-online linear / finite-rank kernel, and projected RKHS truncation. Small-scale numerical experiments illustrate robustness calibration, strategic coverage, and the return-gap trade-off predicted by the framework.
math.NA
We develop a self-supervised operator-learning framework for optimal mixing in incompressible flows, aimed at overcoming the computational limitations of classical PDE-constrained optimal control. The objective is to design velocity fields that efficiently homogenize a passive scalar while minimizing kinetic energy, quantified through the mix-norm. Classical methods solve a separate optimal control problem for each initial condition using iterative forward-adjoint procedures, leading to substantial computational costs and limited scalability. In contrast, we propose to learn the solution operator that maps initial scalar fields to optimal mixing controls directly. The proposed self-supervised method does not require access to supervised labels, thereby significantly reducing the data-generation burden of supervised operator-learning methods. Our framework integrates a Lagrangian reformulation of the transport equation with a transformer-based neural operator architecture. The transport dynamics are represented through characteristic ODEs, enabling particle-based Monte Carlo optimization, while the learned velocity field is parameterized in a divergence-free spectral basis enforcing incompressibility and no-penetration conditions exactly. Numerical experiments on parametric families of initial conditions in two and three spatial dimensions demonstrate robust generalization to unseen regimes and approximately exponential decay of the mix-norm. The learned controls also transfer consistently to independent Eulerian finite-volume simulations, validating the physical fidelity of the learned operator. To the best of our knowledge, this work provides the first self-supervised operator-learning framework for optimal mixing and among the first data-driven approaches for optimal mixing in fully three-dimensional incompressible flows.
cs.AI
Meteorological downscaling is crucial for high-resolution regional climate forecasting and disaster early warning. While neural operators have emerged as a promising paradigm for modeling complex spatiotemporal mappings, existing frameworks often struggle with spherical manifold geometric distortions, inherent atmospheric multi-scale coupling mismatches, and lack of explicit atmospheric laws. We propose the Spherical Physics-informed Neural Operator, which utilizes a Spherical Laplacian Decomposition to partition atmospheric fields into hierarchical frequency components, maintaining exact point-wise correspondence across scales. To evaluate these representations at arbitrary locations, we introduce a localized spherical integral operator that approximates continuous kernel transforms via geometry-aware attention. Dynamical consistency is further enforced by embedding differentiable constraints into the learning process. Extensive experiments demonstrate that our framework attains superior accuracy and zero-shot generalization across various meteorological variables and unseen queries, representing a robust and interpretable solution for global-to-regional meteorological downscaling.
physics.plasm-ph
Real-time control of tokamak plasmas requires the electron density profile. However, in a reactor-relevant device, diagnostic port access is limited and the profile can be observed only partially. In the present work we reconstruct the density profile by inferring the coefficients of a physics-based one-dimensional particle transport equation, using a Fourier Neural Operator (FNO) surrogate of the transport solver as a fast differentiable forward model. The surrogate enables realtime-compatible gradient-based reconstruction and uncertainty quantification from the reflectometry and interferometry measurements. We compare four uncertainty quantification methods - multi-seed ensembles, the Laplace approximation, Metropolis-Hastings, and a Gaussian method independent of the reconstruction fit. The resulting uncertainty is scored with CRPS and coverage calibration, separately on the observed and unobserved regions. We further demonstrate advantage of our approach over standard analytical function fit on synthetic and experimental DIII-D discharges.
math.NA
We introduce a novel neural operator method based on Segal-Bargmann transform and Bargmann-Fock space for learning the solution operators of partial differential equations (PDEs). The method brings data encoding, latent-space representation, operator learning, and decoding into an unifed framework. From only finitely many observations, we establish a sup-norm reconstruction error of order , where L is the number of complex sampling points that are decoupled with the original samplings and observations. The reconstruction error is independent of the spatial dimension and the number of original sampling points. We show that this new mesh-free Segal-Bargmann Neural Operator approach has superior performance compared to existing NO approaches like FNO, BelNet, GNOT, etc.,\ particularly in terms of pointwise control for solutions of PDEs which provides pointwise accuracy and interpretability as classical numerical solvers. We also demonstrates the reconstructions of the solution operators for higher dimensional nonlinear PDEs, which suggest that our proposed method has great potential to be applied to more challenging PDEs and to even higher spatial dimensions.
cs.CV
We present Spatons, a family of compositional, spatially localized neural operators for computationally efficient XR (extended reality) predistortion and distortion compensation. XR displays (AR, VR, and MR) suffer from spatial (e.g., geometric) and tonal (e.g., colour) distortions. Frames can be pre-distorted so that the projected image looks as intended. Distortions vary with manufacturing tolerances and projection systems, motivating programmable compensation. Untethered, battery-operated XR systems also require low energy and latency. A Spaton is a hybrid patch-to-pixel block combining tiny Multi-Layer Perceptrons (MLPs) with fixed-function interpolation. Using Spatons, we build programmable, computationally efficient pipelines for varied distortion scenarios. On the full spatiotonal Scenario 5, Sequential-5' matches CompenNet++ in SSIM (0.944 vs. 0.944) and closely matches its PSNR (27.5 vs. 27.7 dB), with substantially lower complexity. We demonstrate a flexible Spaton solution with 6 times fewer FLOPs and 106 times fewer parameters than CompenNet++, and 4-10 times fewer memory transfers than UNet++. We characterize software implementations and analytical hardware estimates, including optimizations across target XR platforms. The distortions considered are drawn mostly from see-through, near-eye displays, where the spatial light path governs local intensity and spectral response. This early systems-level exploration uses closed-form differentiable distortions and one real projector-camera capture set for training and evaluation, rather than through-the-optics measurements. Testing on a wider range of real XR devices remains future work. Chromatic dispersion (color fringes) remains the main artifact where Spatons trail monolithic baselines.
cs.LG
Neural networks whose inputs are functions, as in operator learning, are usually shown to be universal by existence arguments, either by reduction to finite-dimensional networks or by Stone–Weierstrass-type density; these prove that approximants exist but say little about how the parameters depend on the target. We take a constructive route. We extend the ridgelet transform, an integral transform related to the Fourier, Radon, and wavelet transforms, to continuous maps on a separable infinite-dimensional Hilbert space, replacing the Lebesgue measure that is unavailable in infinite dimensions by Gaussian weights with a homogeneous mixture on the direction space. The resulting transform has a Plancherel identity and reconstruction formulas incorporating the frame operator, the correction induced by the weights, and it assigns to each target an explicit coefficient measure, one for every admissible analysis filter. From these representations we obtain uniform approximation on compact sets for all continuous non-polynomial activations of polynomial growth, including ReLU, and, by Monte Carlo sampling of the coefficient, finite-width networks with a dimension-free error in the width for globally Lipschitz activations and coefficients with finite variation and second moments. Explicit examples include Gaussian targets and neural-operator layers. A companion Lean~4 formalization provides machine-checked proofs, with its correspondence to the manuscript documented at https://shosonoda.github.io/lean-operator-ridgelet/ .
physics.med-ph
Cerebral arterial stenosis alters blood flow throughout the vascular network, complicating the estimation of its location and severity from sparse velocity observations. We developed a conditional neural operator that predicts cerebral hemodynamics for candidate stenoses and enables inverse estimation through waveform comparison. Training data were generated using a sequential 0D-1D model that simulates pressure, cross-sectional area, and velocity under varying anatomical, physiological, and stenosis conditions. The operator encodes a single stenosis within vessel tokens and uses global attention to represent its effects across the network. Severity-gated coupling connects these representations to a shared field decoder, while paired healthy simulations constrain baseline predictions. After training, model weights remain fixed, and stenosis location and severity are estimated by minimizing the mismatch between predicted and observed midpoint velocity waveforms. Network-averaged relative validation errors were 0.14% for pressure, 0.59% for area, and 2.79% for velocity. For an eight-vessel observation configuration, localization accuracy across 20 candidate vessels was 99.4% without added noise and 96.0% at the 30% noise setting. Comparisons between configurations with equal numbers of observed vessels showed that measurement placement influenced recovery. Errors and localization ambiguities were greater in selected communicating and distal arteries. These results demonstrate the feasibility of surrogate-based stenosis estimation in simulated cerebral networks and highlight the importance of observation placement. Evaluation was limited to unseen stenosis scenarios within sampled anatomical and physiological conditions. Generalization to new individuals and clinical measurements requires further investigation.