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
In variable-step operator learning, agreement at observed step sizes does not by itself determine predictions at other temporal resolutions. We propose an operator whose step-size dependence is prescribed by a first-order Duhamel approximation of semilinear dynamics. It combines exact linear propagation with a local nonlinear function of the field and its derivatives, allowing evaluation below the observation interval without supervision densely sampled across step sizes. Under suitable conditions and at fixed spatial resolution, we prove that agreement between models at two step sizes controls their prediction differences throughout a prescribed range. We also give conditions for local identifiability of both components from two step sizes, supporting interpretation of the learned structure. A latent extension applies the same construction to learned fields. We evaluate the base model on two synthetic problems and the extension on two real datasets against several baselines.
cs.AI
Sparse-view computed tomography (CT) reduces the number of acquired projections, but conventional sampling strategies are designed primarily for image reconstruction rather than for downstream physical inference. Here, we investigate optimal task-aware projection selection for directly predicting hemodynamics from sparse CT angiography (CTA) measurements. Using computational fluid dynamics simulations of intracranial aneurysms, we generate synthetic fan-beam CTA projections and train a Fourier neural operator (FNO) to predict two targets: the spatially resolved time-averaged wall shear stress (TAWSS) field and the spatially averaged TAWSS. The FNO was coupled with optimal input selection for simultaneous optimal sampling and training. Our goal was to investigate if agentic AI could discover optimal sampling strategies that outperform current methods. We compared fixed uniform and golden-angle sampling with three task-aware optimal selection strategies: Gumbel-Softmax, a Soft top-K method, and an agentic AI selector. The methods were compared at different view selection budgets. A detailed analysis of the AI agent was performed to understand its reasoning and performance. For spatially resolved TAWSS, Agentic and Soft top-K achieve the lowest mean errors (Agentic performs slightly better) and substantially outperform fixed sampling under extreme sparsity. For spatially averaged TAWSS, optimized selection becomes important at extremely sparse acquisitions. Analysis of the selected masks reveals multiple distinct subsets concentrated around informative regions. Moreover, masks optimized for hemodynamic prediction yield poor image reconstructions, showing that task-optimal acquisition differs fundamentally from reconstruction-optimal acquisition. This work suggests the possibility of designing sparse acquisitions directly around downstream physical quantities.
math.OC
We introduce the Safeguarded Neurally-Accelerated Preconditioning (SNAP) framework for hybridizing nonlinear optimization methods with pre-trained neural operators (NOs). SNAP targets parametric optimization problems arising from variational formulations of nonlinear partial differential equations (PDEs), where many parameter instances require repeated solves. Inspired by multilevel line-search methods, SNAP operates in cycles, combining classical optimization steps with a search direction generated by a pre-trained NO. Since NO-generated search directions carry no a priori descent guarantee, SNAP employs an a posteriori safeguard by accepting only corrections that satisfy prescribed descent and acceptance conditions. We prove global convergence of the SNAP algorithm and show that it preserves the convergence rate of the underlying optimization method independently of the employed NO architecture. Moreover, we also show that when the NO corrections are sufficiently accurate, SNAP converges faster, with the improvement directly related to the NO training error. To demonstrate SNAP's efficiency in practice, we run numerical experiments on a diverse set of nonlinear parametric optimization problems, including parametric minimal surface, Allen–Cahn, and nonlinear Helmholtz problems. The efficiency of SNAP is demonstrated by comparing two NO architectures, Fourier neural operator (FNO) and Transolver, and by using SNAP as a standalone solver and as a nonlinear preconditioner. Our numerical results show a substantial reduction in the number of required iterations, while demonstrating robustness and generalization across a wide range of problem parameters and mesh resolutions.
cs.LG
Full-order surrogates for high-dimensional nonlinear flows are often unnecessary when applications require only selected degrees of freedom. We propose reduced-order models that bypass full-state reconstruction by deploying graph neural networks (GNNs) directly on coarse meshes. Applied to parameterized 2D/3D confined cylinder flows and an unevenly heated cavity over Reynolds/Rayleigh-number ranges, the models generalize to unseen conditions and accurately predict sparse probe locations and integrated quantities. Coarse-mesh GNNs match or outperform finer-mesh surrogates while reducing training time and memory, even for 3D chaotic dynamics. A dynamics-informed adaptive sampling strategy guided by frequency-spectrum variations reduces maximum errors by an order of magnitude versus uniform sampling at negligible cost. Coarse-mesh training also makes noise-augmented multi-step unrolling practical, improving long-horizon autoregressive predictions. The approach outperforms parametric DMD, sparse MLPs, U-Nets, and Fourier neural operators at lower cost, and enables system identification and state estimation from sparse observations. These results show that coarse-mesh representations, adaptive sampling, and noise-augmented unrolled training form an efficient framework for reduced-order modeling of parameterized unsteady flows without solver-resolution reconstruction.
cs.CE
We study whether the equilibrium of a drop on a chemically patterned wall, and the capillary forces that the pattern exerts on it, can be learned from the free energy alone, without solution data. We introduce DropFENO (Droplet Free-Energy Neural Operator), in which Fourier (FNO), tensor-factorised Fourier (TFNO) and wavelet (WNO) neural operators are trained in two and three dimensions on the diffuse-interface free energy of the drop, with its volume and centroid constraints imposed as penalties. The predictions are tested with the virtual-work (Hellmann–Feynman) theorem: the force exerted by the pattern on a held drop is a wall integral of the equilibrium field which, for a drop straddling a wettability step, reduces identically to the contact-line law . DropFENO reproduces this law and the energy landscape of a drop crossing a step, and reproduces the pinning of the receding contact line less accurately. The residual error lies in the soft modes of the capillary energy (small displacements of the interface and the contact line). This explains why the energy is predicted accurately while contact-line positions are not, and why on striped walls the operators return Cassie-like states instead of the edge-pinned states whose anisotropic apparent angles the reference minimiser shares with lattice Boltzmann simulations. On held-out patterned walls the median energy of the predictions lies $\NdEmedRandFnoTwo$ of the free-drop energy above the best known minimum (FNO, two dimensions), at about one hundredth of the cost of a per-instance minimisation, and a refinement by a classical solver started from the prediction returns the reference state in almost all cases.
math.NA
Parametric PDEs in science and engineering are often posed on irregular meshes with geometries that may contain holes or complex boundaries, yet popular operator-surrogate architectures such as the Fourier Neural Operator (FNO) rely on structured geometries for fast computational kernels. We introduce Adaptive Modal Operator Refinement (AMOR), which replaces FNO's Fourier-based spectral convolutions with reduced-basis blocks and supports both data-driven and prescribed input/output bases through a common encode/decode interface. AMOR thus applies unmodified to arbitrary meshes and, because its spectral parameterization has no tensor-product dependence on the spatial dimension, it scales favorably to three dimensions, where FNO's spectral weights grow as with the number of spectral modes . We prove that AMOR is a universal approximator of continuous operators. On two-dimensional structured-grid benchmarks, AMOR is competitive with FNO overall and surpasses PCANet: it outperforms FNO on nonlinear diffusion and Helmholtz problems, while a static snapshot basis keeps it from matching FNO's accuracy on the time-dependent Kolmogorov flow problem. On three-dimensional problems it matches or exceeds FNO's accuracy at lower inference cost on linear elastostatics while trailing FNO on a turbulent lid-driven-cavity flow and scales favorably to problems demanding high capacity. On irregular meshes, where FNO is not directly applicable, AMOR reduces the relative error over the FNO extension Geo-FNO by (airfoil) and (elasticity). Finally, AMOR maintains strong performance across three physically distinct, experimentally motivated loading regimes on a topologically nontrivial three-dimensional aircraft-rib mesh, demonstrating its applicability to realistic engineering problems without modification.
math.NA
We develop a spectral Barron regularity framework for nonlinear operators between infinite-dimensional function spaces. Using Hilbert-valued Fourier representations of finite-dimensional input truncations, we establish bounds that are uniform in the input truncation dimension and prove arbitrary-order Barron regularity for a class of elliptic solution operators. Second, we use this operator Barron regularity to derive approximation error bounds for an FNO-type neural operator, with explicit dependence on the input truncation dimension, output reconstruction dimension, and network width. For a fixed input truncation dimension, optimizing the neural approximation bound over the Barron regularity order yields stretched-exponential decay of the approximation error in the network width. We also show some numerical experiments to support the predicted rate.
cs.IT
Fluid antenna systems (FAS) leverage continuous reconfigurable apertures to transcend the spatial limits of fixed-position arrays, yet acquiring full-aperture channel state information (CSI) from sparse observations remains a formidable bottleneck. In this paper, we establish a continuous-field formulation for FAS by recasting channel estimation as a stochastic inverse field-reconstruction problem constrained by the projected Helmholtz model. Through electromagnetic information theory (EIT)-based spectral leakage analysis, we demonstrate that finite-aperture reconstruction induces an intrinsic convolutional relationship in the spectral domain, rendering the estimation task a fundamental frequency-domain inverse deconvolution problem. To solve this, we propose a U-shaped Fourier neural operator (UFNO) specifically tailored for noisy, underdetermined wireless environments. Furthermore, we leverage the Landau–Pollak theorem to derive an aperture-dependent EDoF bound that guides the spatial mode budget, and employ a physics-aware Sobolev loss to promote consistency in both channel values and spatial variations. Unlike conventional learning-based approaches, a single frozen UFNO can be directly evaluated at unseen spatial–frequency resolutions without resolution-specific retraining or explicit output interpolation. Extensive simulations show that our framework significantly outperforms state-of-the-art benchmarks in accuracy while achieving orders-of-magnitude lower inference latency than iterative solvers at high spatial–frequency resolutions.
physics.flu-dyn
The standing wave problem concerns the space-time periodic motion at the free surface of a two-dimensional, infinitely deep, irrotational flow of an incompressible inviscid fluid under gravity. Although spectrally accurate computations have revealed a rich array of nonlinear wave phenomena, repeatedly evaluating their evolution over large parameter spaces remains challenging, particularly when resolving fine-scale spatial oscillations. We employ a Fourier neural operator (FNO) architecture to learn the solution operator along a numerically computed one-parameter family of symmetric standing waves. Given a wave profile at an instant of rest and a normalized observation time, the learned operator predicts the fluid surface, the velocity potential and the dipole density at the surface at that time. In-distribution tests yield uniformly small relative errors within the training range. Two out-of-distribution experiments characterize extrapolation toward contrasting physical regimes. Toward the free-fall limit, errors concentrate near the sharpening crest, consistent with the spectral truncation in the FNO architecture. Toward the linear limit, absolute errors remain small whereas relative errors increase as the amplitude vanishes. A residual of the governing equations, evaluated independently of reference solutions, complements these tests and provides a diagnostic for exploratory predictions beyond the available data.
math.AP
In this paper, we study neural operators for the Dirichet-to-Neumann map (DN map), which is encountered in the famous Calderón's problem. In many numerical methods used for reconstructing the unknown conductivity from the knowledge of the DN map need to evaluate the corresponding DN map at a given conductivity function, which involves solving the boundary value problem for the conductivity equation and taking the Neumann data. Such procedure is time consuming and not effective. Hence, it is a favorable strategy to construct a surrogate to replace the forward map mapping conductivity functions to corresponding DN maps. The main theme of this paper is to derive quantitative bounds on the parametric complexity of these neural operators. In particular, under suitable assumptions, their depth grows at most logarithmically with , while the relevant spectral and projection ranks grow algebraically. It should be noted that general neural operators suffer from "the curse of parametric dimensionality". This work demonstrates the construction of neural operators for the DN map that does not suffer from such curse.
cs.LG
The Well, a 15-terabyte dataset spanning 16 physical simulation domains, provides an extensive benchmark for developing and evaluating machine learning surrogate models for complex physical systems. The baselines released with the benchmark were presented by their authors as illustrative examples obtained at small computational cost, trained with zero to very limited hyperparameter tuning and under a highly constraining 12-hour per-run wall-clock limit. These results have subsequently been treated as fully tuned benchmarks, and in some cases effectively interpreted as unconstrained performance baselines. We perform a short non-exhaustive hyperparameter search for the Tucker-factorized Fourier Neural Operator (TFNO) under the same 12-hour per-run wall-clock limit, while using only a single input timestep rather than the history of up to four used by the published baselines. Despite this more restrictive setup, tuning alone reduces VRMSE below both published FNO and TFNO baselines on 15 of 17 tasks, sometimes by several orders of magnitude. This sensitivity to a modest tuning effort underscores that the released neural operator baselines should not be treated as architectural benchmarks.
cs.AI
Earth-system models simplify atmospheric chemistry into condensed mechanisms, because near-explicit mechanisms such as the Master Chemical Mechanism (MCM) are too costly to solve in every grid cell. Neural emulators could lift this limit, but most advance the chemistry in small time steps and cover only a few species or a reduced mechanism. Here we introduce FNO-MoE, a Fourier neural operator with a mixture of experts that predicts the full near-explicit chemical state over 30 minutes in a single call. The same network also guides its own training, using the disagreement among its experts to select the most informative box-model simulations. At equal cost, this active sampling yields a more accurate emulator than random or evenly spread sampling. Fine-tuning on longer trajectories keeps it stable over multi-hour runs in which models trained on short simulations diverge. The resulting emulator reproduces the evolution of thousands of species with high fidelity, remains accurate for unseen chemical compositions and runs orders of magnitude faster than the numerical solver. These results open a new direction for emulating complex atmospheric systems: neural operators that predict the full chemical state over long windows and choose their own training data. Training under changing meteorology is the next step towards bringing near-explicit chemistry into Earth-system models.
cs.LG
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.
cs.CV
Automotive millimeter-wave (mmWave) radar has attracted increasing attention for semantic segmentation in autonomous driving, owing to its robustness under adverse weather and lighting conditions. Recent works have introduced radar-specific neural operators such as PeakConv (PKC), inspired by constant false alarm rate (CFAR) detection, to better model the peak-dominated characteristics of radar images. However, existing methods typically treat the guard bandwidth as learnable parameters, without explicitly characterizing the frequency response of the PKC operator. In this work, we revisit PKC from a frequency-domain perspective and propose Frequency-Aware PeakConv (FreqPKC), which explicitly models and exploits its underlying spectral characteristics. By analyzing the frequency response of PKC, FreqPKC enables the network to capture a broader range of frequency components and adaptively adjust its effective bandwidth. With the proposed Frequency-Aware (FA) module, the network can infer appropriate guard bandwidths to balance high- and low-frequency information: it enhances high-frequency components in target regions for more precise segmentation while suppressing noise through low-frequency filtering in background areas. Building on FreqPKC, we further develop FreqPKC-Net, a radar semantic segmentation (RSS) framework that integrates frequency-guided adaptive modulation with cross-view fusion for multi-view inputs, and can also be adapted to single-view data. Experiments on the public datasets CARRADA, CARRADA-RAC, and a self-built dataset named RS-SAR demonstrate that FreqPKC-Net consistently outperforms existing SoTA methods on both range-Doppler and range-angle views.
cs.LG
Decisions about heat, frost and flooding are made locally, yet reanalyses and climate models resolve tens of kilometres. Generative downscalers add kilometre-scale detail with uncertainty, but are tied to a grid, costly to sample and lack a tractable likelihood. Here we introduce CD-FFM, a functional flow-matching model that treats the high-resolution field as a continuous random field that one model can query at any location and resolution. Its velocity acts pointwise, so spatial variability comes from a noise prior fitted to the observed spectrum and every observation has an exact likelihood. Downscaling ERA5 to 1 km daily temperature over Fennoscandia, CD-FFM lowers the continuous ranked probability score by 27-31% against regression and 12-18% against a forty-times larger neural operator in each of three test years, and outperforms a CorrDiff-style diffusion model while sampling about fifty times faster. The same model transfers to precipitation and makes large kilometre-scale ensembles affordable.
cs.LG
Computational Fluid Dynamics (CFD) is essential in contemporary engineering design, yet it remains expensive to compute and can take thousands of CPU/GPU hours or more for a single high-fidelity simulation. Recent developments in machine learning for scientific modeling have led to powerful surrogate frameworks that can approximate flow fields and physical quantities by using a fraction of the computational cost. However, most of these methods require largescale homogeneous datasets or simple geometries, exhibiting severe generalization gaps when applied to complex and data-scarce configurations. To address this, we propose the Factorized Geometry Convolutional Neural Operator (FG-ConvNO) for robust geometry-aware propeller pressure prediction under limited training data. We introduce geometryaware encoding, efficient interpolation-based decoding, and auxiliary global quantity supervision. In addition, we incorporate DISCO blocks that use discrete continuous convolutions to make the model discretization agnostic. In our experiments, FG-ConvNO achieves the highest accuracy for both surface pressure and global thrust prediction on a procedurally generated propeller CFD dataset. Geometry-aware encoding and architectural simplification improve stability, while random vertex subsampling enhances robustness. Overall, adding geometric priors to grid-based convolutional operators reduces overfitting on complex propeller geometries and improves generalization in low-data regimes.
cs.LG
State inference and parameter learning for nonlinear filters is a core computation for scientific machine learning with varied applications in engineering and neuroscience. The unnormalized filtering distribution is governed by the Zakai equation, a stochastic partial differential equation (SPDE) that combines the forward Kolmogorov operator (known as the Fokker-Planck equation) with observed likelihood updates. Although solving these high-dimensional SPDEs is often impossible analytically and intractable numerically, advances in operator learning and physics-informed machine learning have made it feasible. Harnessing these advances, we introduce the Zakai Equation Neural Operator (ZENO), a model that optimizes a neural operator against a Zakai Equation based physics-informed loss to approximate the exact unnormalized filtering distribution for latent stochastic dynamical systems. We use this ZENO-inferred posterior distribution in an expectation-maximization (EM) framework to perform parameter learning and state inference. We demonstrate ZENO's ability to accurately infer the posterior distribution and learn its corresponding SDE parameters in three simulated stochastic dynamical systems (Double-Well bistable system, Van Der Pol (VDP) oscillator, and Lorenz system) and compare its performance against established methods for latent SDE inference and learning. We show that ZENO recovers previously observed limit cycles representing forward and backward locomotion from a whole-brain imaging dataset of immobilized \textit{C. elegans} nematodes. ZENO is a novel approach to the problem of nonlinear filtering and dynamical inference through the lens of partial differential equations and operator theory.
physics.comp-ph
High-fidelity Reynolds-averaged Navier–Stokes (RANS) simulations remain a standard tool for aerodynamic analysis, yet each solve on an unstructured mesh is computationally expensive, making many-query tasks such as design exploration, optimization, and multi-condition evaluation costly. Neural operators provide mesh-flexible surrogates for mapping geometry and operating conditions to flow fields; however, in many existing architectures, global parameters such as the freestream state and design variables are concatenated with pointwise inputs, leaving their influence on intermediate features implicit. This paper proposes CoPiT, a \textbf{Co}nditional \textbf{P}osition-induced \textbf{T}ransformer that builds on the Position-induced Transformer (PiT) as its geometric backbone and injects a global condition vector through multi-stage feature-wise linear modulation (FiLM), thereby making parametric modulation explicit while preserving geometry-based attention weights. The model is evaluated on two-dimensional AirfRANS airfoil fields, including scarce-data and extrapolation regimes, and on three-dimensional BlendedNet++ blended-wing-body surfaces. On the AirfRANS full-data test set, CoPiT reduces surface and volume losses by approximately 54% and 24% relative to Transolver, respectively, and achieves the lowest overall normalized test MSE among the compared baselines on BlendedNet++. Matched ablations show that FiLM conditioning generally improves upon input concatenation, while the preferred placement is task-dependent. CoPiT thus provides a practical operator-learning surrogate for parametric aerodynamics on unstructured meshes.
physics.flu-dyn
Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier–Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.
stat.ML
Offline maximum entropy reinforcement learning seeks to estimate the optimal soft action-value function \(Q^*\) from a fixed dataset without further interaction with the environment. Although \(Q^*\) is characterized by the soft optimal Bellman equation, the corresponding Bellman operator cannot be evaluated directly because both the reward distribution and the transition kernel are unknown. Consequently, estimating \(Q^*\) requires the joint treatment of environment modeling and value function learning. We propose a framework that addresses these tasks separately. Conditional diffusion models based on probability flow ODEs estimate the conditional reward distribution and transition kernel. The resulting distributional estimators define a data driven approximation of the soft optimal Bellman operator, whose empirical Bellman residual is minimized over a class of deep neural networks to estimate \(Q^*\). We establish nonasymptotic guarantees for the proposed method. For operator learning, we derive total variation error bounds for the conditional flow estimators that account explicitly for score estimation, numerical discretization, and terminal smoothing. For soft value learning, we first use the soft Bellman equation to show that inherits Hölder regularity from the conditional reward and transition densities. For any Hölder index , the value learning stage achieves an excess Bellman residual risk bound of order where is the total dimension of the state and action spaces. In contrast to many existing theoretical analyses of deep reinforcement learning, our results do not require Bellman completeness. Numerical experiments support the theoretical findings and demonstrate the effectiveness of the proposed method.