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
The penetration-resistant design of concrete structures requires the spatiotemporal evolution of the stress field within the target, whereas in design only the strength grade, the geometry and the projectile parameters are known, and the aggregate arrangement is not. Empirical formulae and scalar-output surrogates serve design screening, and mesoscale finite element models resolve the aggregate-scale response, but one solution cannot cover random aggregate samples at scale, and no framework links mesoscale heterogeneity to field-level prediction. A physics-coupled graph neural operator (PC-GNO) is proposed, which learns in function space the evolution of the mid-plane six-component stress tensor field. The training data are generated by full-scale, aggregate-resolved finite element models and comprise 359 cases at the impact velocities of 100, 135 and 165 m/s, corresponding to compressive strengths of 14, 23 and 48 MPa; the data source is verified by laboratory measurements. The coupling strength of message passing is constructed from the impedance contrast across the material interface, and the update is constrained by spectral normalization. Under the single-step primary setting, the mean accuracy of the three levels is 80.16% (91.45%, 77.80% and 71.22%), i.e. 27.81, 42.46 and 34.16 percentage points above the preceding-frame field of the same window; with 20 664 parameters it exceeds the 87.9% of the best Fourier neural operator configuration (about 812 times as many parameters) and the 71.0% of the closed-form linear regression. The output is the complete six-component tensor field rather than a fixed engineering quantity, so that strength criteria and energy indicators follow from one set of weights. For field-level response prediction in concrete penetration, this result provides a data-driven route, with conditions of applicability defined by the training and the evaluation.
cs.LG
Guided sampling with sparse observations and PDE residuals has recently been combined with neural operator priors to solve partial differential equations in function space, their natural setting. We pair such a prior with pseudo-bootstrap sequential Monte Carlo (pBS SMC), a guided sampling framework that accommodates proposals without tractable densities, and introduce a new proposal by adapting the DPM++ algorithm to the guidance setting. Across five benchmark PDE systems this proposal attains the best error–time trade-off at every step count tested, and wrapping it in pBS gives the lowest error on most inverse-problem settings, with the largest gains when few sampling steps are used.
cs.LG
Morphogenesis is driven by the interplay of genetic regulation and tissue mechanics. In vivo video data is limited, particularly under genetic mutations and environmental perturbations, making generalization beyond the most representative conditions in the dataset challenging. However, at tissue scales, the dynamics of Drosophila can be described as a continuum flow, motivating the usage of representations learned across other physical systems as priors for developmental dynamics. We investigate finetuning Walrus, a Continuum Dynamics Foundation Model pretrained across a broad set of simulated continuous systems, on tissue velocity fields from only 12 wild-type Drosophila blastoderm videos. We evaluate autoregressive prediction both in-distribution and zero-shot under unseen temperature and genetic perturbations. Zero-shot refers to evaluation on perturbation conditions never seen during wild-type finetuning, without any additional training. In-distribution, a Fourier Neural Operator model trained from scratch achieves best performance on VRMSE. However, Walrus (CRPS) is the strongest model under the most severe distribution shift of unseen samples of mutated embryos and temperature conditions. Interestingly, predictions capture the shape of the mean surface tissue velocity trajectory and its separation between 17°C and 27°C, including the timing of the peak at Germ Band Extension onset, a biologically meaningful milestone of embryo development. Our results provide preliminary support that representations learned from different dynamical systems can transfer to experimental morphogenesis in developmental regimes, provided they admit a continuum description. More broadly, our results motivate further study of physics foundation models as a new complementary tool for the simulation of morphogenesis in low-data regimes.
math.NA
We propose a conservative local neural operator (CLNO) for accurate and generalizable autoregressive prediction of compressible flows. Building upon the locality and translation invariance of the local neural operator (LNO), CLNO introduces a conservative evolution framework based on local subdomains of fixed physical size, separating local mean-state evolution from fine-scale reconstruction. The flow is encoded into subdomain-integrated conservative states and fine-scale features, while a shared local network predicts interface fluxes for finite-volume updates that enforce discrete conservation. A convolutional reconstruction network recovers fine-scale structures using zero-mean basis functions, preserving the updated local means and providing information for subsequent evolution. To accommodate different physical parameters, Reynolds–Mach number conditioning modulates the encoding, flux-prediction, and reconstruction networks, enabling a single model to learn parameter-dependent dynamics and generalize to unseen parameter combinations without retraining. CLNO supports prediction across spatial resolutions, computational domain sizes, and boundary conditions. Numerical experiments demonstrate improved generalization and competitive or higher accuracy compared with LNO, including order-of-magnitude error reductions in representative cases, at approximately three to five times the inference cost on periodic benchmarks. These results demonstrate accurate, generalizable, and physically consistent flow prediction through conservative local evolution and fine-scale reconstruction.
cs.LG
Learning solution operators for partial differential equations (PDEs) requires models that can represent spatial structure across grids and scales. This becomes especially important in time-dependent problems, where the learned operator is often used autoregressively: each predicted state is fed back as input to predict future states. In this setting, small spatial errors can be repeatedly propagated through the learned dynamics, leading to instability and drift, vitiating long-term solution fidelity. We propose Mesh-Free Convolutions (MFCs), filters defined in continuous space and discretized only at evaluation time. This allows the same learned filters to be applied across resolutions. Rather than imposing a grid-dependent stencil or a fixed spectral cutoff, MFCs learn smooth scale-dependent filtering and include directional terms for transport-like behavior. Across fluid-dynamics and weather-forecasting benchmarks, MFC-based models produce stable long-horizon rollouts, transfer reliably across resolutions, and better preserve long-time statistics. These results suggest that continuous, mesh-free filtering is a useful inductive bias for PDE operator learning, especially in autoregressive prediction.
eess.SY
Fourier neural operators predict power-system transient trajectories with high accuracy. We extend this line of work by examining the choice of spatial spectral basis, studying both the graph Fourier transform derived from the grid's own coupling network and learnable bases trained from that starting point and from a Fourier control. Evaluated in one controlled pipeline across three protocols spanning unseen fault locations and shifted operating regimes, the learnable bases improve on a parameter- and budget-matched DFT variant by 7-14% in distribution, with per-case significance on both in-distribution protocols and no loss on the third; the gain comes from adaptation rather than from the coordinates alone. We further characterize the regime dependence of this gain, identifying its cause, locus, and continuity, and show that a resolution requires no new model: averaging the learned and frozen variants gives the best shift-robust predictor, while their disagreement provides a label-free alarm that flags every shifted case with no false alarms.
cs.LG
Full-waveform inversion (FWI) is a powerful approach for reconstructing subsurface velocity models, but its practical performance is limited by ill-conditioning, sensitivity to the update direction, and the high cost of curvature-based methods. In this project, we investigate whether a neural operator trained on reverse-time-migration (RTM) images can provide a useful learned preconditioner for FWI. We compare projected FWI against neural-preconditioned FWI using either a standard Fourier neural operator (FNO) or a Jacobian-informed FNO (JI-FNO). The neural preconditioner applies the normal operator of the learned RTM map to modify the true Devito FWI gradient. Our experiments show that the JI-FNO preconditioner achieves the lowest model error while maintaining data-residual decay comparable to baseline projected FWI. In contrast, the standard FNO preconditioner is unstable and gives worse model recovery, despite using the same neural-preconditioning framework. We also identify a failure mode of a nonlinear neural Gauss Newton-type inner solve: directly matching the neural RTM output to the FWI gradient creates an objective-space mismatch and does not yield a reliable descent direction. These results suggest that forward RTM-image prediction alone is insufficient for learned preconditioning; accurate learned derivatives and objective-consistent use of the neural operator are critical for producing useful inversion updates.
math.NA
Conventional control approaches to optimization problems under uncertainty entail a practical compromise: deterministic controls offer robustness at the expense of pathwise accuracy while stochastic controls can adapt to individual realizations but suffer from the curse of dimensionality. To achieve both robustness and adaptivity, a neural operator-based approach is proposed in this paper for distributed elliptic optimal control problems with high-dimensional random inputs. A data-driven training algorithm is first established to learn the mapping from coefficient realizations directly to their optimal pathwise controls. To bypass the burden of generating training data pairs, a physics-informed training algorithm is then proposed and recast as weakly supervised learning using pseudo label targets produced by standard adjoint-based control updates. Numerical experiments with rough coefficients are conducted to validate the effectiveness and efficiency of our robust-pathwise controls, which outperforms the benchmark robust deterministic control on unseen coefficient realizations owing to the generalization capability and fast inference of trained models.
physics.acc-ph
Particle-in-cell (PIC) simulation is indispensable for modeling beam dynamics in particle accelerators, but its computational cost limits many-query workflows, such as configuration exploration, virtual diagnostics, and controller design. We introduce GLOBE, a physics-constrained Generative Latent Operator for Beam Evolution, modeling the nonlinear, control-dependent, multiscale evolution of 3D beam–field dynamics covering spatial cells. GLOBE compresses beam–field states into a compact latent space and factors their evolution into two components. A control-conditioned neural operator learns the dominant dynamics, while an amplitude-normalized generative model captures the residual distribution, recovering fine-scale structures and conditional variations. GLOBE enforces field constraints by regime-specific, parameter-free operators: a Poisson solve reconstructs the electric field in the electrostatic regime, and a spectral projection imposes Gauss's law and a divergence-free magnetic field in the electromagnetic regime. On two high-resolution 3D particle-accelerator benchmarks, bunched-beam transport and plasma-wakefield acceleration, GLOBE lowers relative- error by 9% and 13%, low-frequency spectral error by 69% and 51%, and mid-frequency spectral error by 54% and 20% relative to the strongest baseline (DPOT), satisfies field constraints, and reduces hours of PIC simulation to a minute per rollout.
cs.LG
Diffusion models typically learn the generative dynamics of a single target distribution or a family of distributions indexed by conditioning variables fixed during training. This limits their ability to adapt to a previously unseen distribution from only a small set of examples. We introduce an in-context solution operator learning framework for diffusion models, in which each target distribution is treated as a task drawn from a meta-distribution and is specified at inference time by an empirical context set. Our method learns an in-context score operator that maps the context samples, a noisy query point, and the diffusion time to the task-dependent score, enabling reverse-diffusion sampling on unseen tasks without retraining or parameter updates. A key observation is that the task-dependent score admits a Gaussian-kernel conditional-mean representation. Its empirical Nadaraya–Watson approximation can be represented exactly by a model in our constructed transformer class, providing a principled foundation for the proposed query-dependent attention architecture. The operator is trained through amortized denoising score matching across task distributions. We establish generalization-error bounds that quantify the effects of the context size, the numbers of training tasks and within-task samples, model complexity, and intrinsic and ambient dimensions. Numerically, we evaluate the method on a range of scientific and visual datasets, including Navier-Stokes, ERA5 climate data, QuickDraw sketches, and Food-101. The results show that the proposed method performs well on relatively simple task families, while its performance degrades as task complexity increases, thereby illustrating both the promise and the limitations of in-context learning for diffusion models.
cs.LG
Partial differential equation (PDE) surrogate modeling often relies on either dense observations or access to governing equations/physics constraints. In practice, observations can be sparse and partial, and the underlying physics may be partially specified or unavailable. We introduce SBNO, an equation-free paired-data diffusion bridge matching framework that learns conditional operators for both forward and inverse PDE problems under partial observations. SBNO uses diffusion Schrödinger bridge matching to learn a pair of time-dependent drifts from a mask-aware endpoint coupling that preserves empirical input-solution pairing during training and supports both PDE-task directions within a single framework. At inference time, SBNO produces a deterministic point reconstruction by integrating the probability-flow ODE constructed from the learned drift pair, without inference-time observation or PDE guidance-weight tuning. On five PDE benchmarks with 5% observations, SBNO achieves superior or comparable accuracy to baselines while demonstrating lower test-instance error dispersion. Furthermore, SBNO is at least faster and reduces GPU memory by at least compared to guided-diffusion baselines.
cs.LG
Neural surrogates provide computationally efficient auto-regressive modeling of wave propagation, enabling e.g. fast inverse solutions. Despite recent progress, the current surrogates are not yet sufficient for the needs of ultrasonic sensing, characterized by high frequency and heterogeneous propagation media. We show how methods building on Fourier Neural Operators (FNO) fail to capture essential signal features, such as reflections, whereas U-Net -based methods learn the wave fields well but lack the ability to transfer information across different spatial scales. We then propose a new surrogate that captures the best of the two worlds, by integrating FNOs and local-integral resampling into a U-Net architecture. We show it can be trained on limited data, and that the predictions are robust to input noise and work across spatial scales.
cs.LG
Neural operators that learn physical laws from data should generalize beyond their training distribution and expose the learned law in closed form. The Kolmogorov-Arnold neural operator achieves both for position-dependent dynamics by learning a dense pseudo-differential symbol over all positions and frequencies, at a cost that grows with the square of the resolution and with closed forms that must be read out by hand. This dense symbol is unnecessary: a finite-order variable-coefficient differential operator has a symbol whose separation rank is bounded by the number of its coefficient functions, independently of the resolution, and the symbols that this network learns on its own benchmarks have rank two. We introduce the Kolmogorov-Arnold flow (KAFlow), which learns the generator of a physical operator as a low-rank symbol built from univariate Kolmogorov-Arnold functions, fixes the gauge of these functions so that every term has a unique place, integrates the dynamics with exact exponential integrators, and distills its closed form automatically. On the synthetic benchmarks of the Kolmogorov-Arnold neural operator, KAFlow lowers the out-of-distribution loss by 3.4 to 11 times and recovers every coefficient to within 10^-4 without symbolic templates, reaching the precision floor of the data. On long-horizon quantum dynamics, it lowers the state infidelity by up to 6 orders of magnitude, and by 30 times when trained only on position and momentum measurements. KAFlow also learns a variable-order fractional operator 15 times more accurately than existing operator networks and discovers nonlocal, fractional and non-polynomial dispersion relations on which sparse regression fails.
physics.comp-ph
Stiff chemical kinetics is a major computational bottleneck in reacting-flow simulations, while data-driven solvers remain sensitive to training-state coverage and chemical dynamics. This work combines conditional diffusion sampling with a thermochemically constrained multiscale Laplace neural operator (TC-MLNO) for chemistry ODE integration. The diffusion model learns the equivalence-ratio-conditioned joint distribution of temperature and species from low-dimensional flame data, augmenting the state space without independent perturbations. TC-MLNO represents transient, non-periodic responses through pole-residue Laplace layers and assigns fast, medium, and slow species to separate branches. Thermochemical constraints are imposed through losses for mass and elemental conservation and heat release rate. The framework is evaluated in 1D premixed flames, a 2D developing jet, and a 3D turbulent burner flame. With equal-sized datasets, diffusion sampling reduces the 95th-percentile distance to the base flame manifold by up to 37.5% and the a posteriori full-species macro-average error by up to 92.7% versus random sampling, especially for minor species and radicals. TC-LNO yields smaller errors than the DeepONet baseline for 84.9% of retained modes and reduces the 95th-percentile error for the most attenuated modes from 0.372 to 0.081. In multidimensional flames, TC-MLNO better preserves reaction-zone locations, heat-release structures, and chemically sensitive species than DeepONet. Errors of representative radical and nitrogen-containing species in the 3D flame decrease by 88.0%-88.9%. Compared with direct CVODE integration, TC-MLNO reduces mean chemistry and total solution times by 90.1% and 62.6%, respectively. These results show that distribution-aware sampling and Laplace neural operator learning provide an accurate and efficient alternative for stiff chemical ODE integration.
astro-ph.EP
Edge-on protoplanetary disks provide a spatially resolved view of the vertical distribution of icy planetary building blocks, but inferring their relative abundances requires radiative-transfer (RT) modeling with multiple anisotropic scattering. Each model costs - CPU hours, which precludes principled Bayesian inference and homogeneous, population-scale analyses. We develop and validate a fast and reusable surrogate for Bayesian inference with ice-bearing edge-on disk models. We train a Fourier Neural Operator on a grid of RADMC-3D spectra. The network maps disk parameters and analytic physical input fields to the spectrum, using mixed-resolution spectra. We use the surrogate in an explicit-likelihood nested sampler and validate it on held-out simulations and synthetic recovery tests with TARP coverage. Finally, we use it to retrieve the disk structure and ice abundances of ESO-H 569 from its spectral energy distribution. The surrogate reproduces the RADMC-3D spectra with a median error of 4.84%. We recover synthetic parameters without detectable bias and with nominal coverage. For ESO-H 569, the posterior favors a massive (), nearly edge-on ( deg) disk consistent with the image-plus-SED model of Wolff et al. (2017). Its ice abundances and cavity size remain weakly constrained by broadband data alone. The marginal spectrum-evaluation speed-up is -. Neural-operator emulation makes full posterior inference feasible within the adopted model and training prior. Key posterior points should still be checked with high-fidelity RT models. By drastically cutting evaluation time, this approach enables for the first time homogeneous population studies of ices in disks, which new JWST surveys can capitalize on to map the range of compositions within planetary nurseries.
cs.AI
Do LLM scientific agents rely only on their initial task context, or do they adapt their decisions in response to experimental feedback? We study this question in neural operator adaptation, where a large language model (LLM) selects fine-tuning configurations under a limited trial budget. Across transfers within and between partial differential equation (PDE) families, the LLM achieves lower held-out test nRMSE than random search and Bayesian optimisation in nearly every matched comparison. Endpoint performance alone cannot distinguish what happens, so we verify each attribution with controlled interventions. Before observing any validation score, the LLM's first configuration already ranks near the top of the corresponding random-search pool, indicating a useful initial bias. A complementary cold-start intervention shows that the selected base learning rate shifts with the PDE description. Once feedback becomes available, reassigning validation scores among evaluated configurations changes the next proposal in every case tested, whereas a value-preserving rewrite produces no comparable aggregate effect. These interventions establish that the LLM's decision-level actions respond to the given task and observed outcomes, showing that it combines a task-dependent prior with sensitivity to experimental feedback.
math.NA
Structure-preserving numerical methods for gradient flows have been extensively developed when the governing equations and free energies are known. However, accurately predicting dynamics from observational data while preserving intrinsic physical properties remains challenging. Although neural operators, such as Fourier neural operators (FNOs), provide powerful data-driven approximations, their predictions do not inherently guarantee physical structure preservation. To address this challenge, we integrate deep learning with structure-preserving exponential time differencing (ETD) methods. Instead of directly learning the evolution operator, we introduce a *trajectory-inferred free energy* (TIFE), which reconstructs the unknown energy density from observed trajectories through a Duhamel-based learning framework. The learned energy determines the variational force and stabilization parameter, enabling structure-preserving numerical evolution. We further extend the framework to jointly identify the free energy and diffusion coefficient. Under suitable assumptions, we establish the maximum-bound principle, unconditional dissipation of the original learned energy, and error estimates incorporating discretization and inference errors. Numerical experiments demonstrate accurate energy recovery and reliable long-time predictions.
math.FA
We study uniform approximation for operator learning on compact subsets of Banach spaces. We bound the recovery dimensions and the number of network parameters in terms of compact-set complexity, the target gauge, and recovery stability. For bounded linear operators with a separable domain, we characterize gauge approximability by coordinate decompositions with a common vanishing gauge and arbitrarily small additive errors. Their partial sums give a fixed family of nonlinear encoder–decoder pairs. We prove that nonlinear recovery with a common vanishing gauge and a rank bound depending only on the model dimension is equivalent to the uniform approximation property (UAP). We also obtain linear recovery estimates with the same rank bound and an operator version for the linear factors of nonlinear targets. For Barron operators taking values in a Banach space of type and admitting a Bochner-integrable density, sampling gives the rate for the expected error, where is a probability measure on the input compact set. For fixed finite-dimensional factors and a class of Hölder maps, we obtain matching upper and lower parameter bounds at fixed sufficiently large depth. Classical sampling and truncation examples provide explicit recovery maps, and a nonlinear integral operator illustrates the parameter bounds.
math.NA
Operator learning aims to learn mappings between function spaces and has been widely used for PDE-related problems. This raises two fundamental questions. First, is there a connection between standard numerical methods and neural operators, given that both are designed to approximate PDE solution operators? Second, when a neural operator is trained on numerically generated data, does it approximate the underlying infinite dimension operator, the training data themselves, or the discretized numerical scheme used to generate them? A key hypothesis in the paper is that the trained neural operator may mimic the discretized numerical schemes that generate the training data. The hypothesis has been partially and preliminary verified by the linear approximation principle proposed in Chen Chen 1995 paper and the following scaling law papers. However, neural operator approximation from the perspective of numerical discretization remains unexplored. In this work, we consider a class of PDEs and derive neural operator architectures from the numerical schemes. Firstly, the numerical basis functions, such as finite element basis functions, can be approximated or directly constructed by learnable neural networks. Then the finite difference schemes for temporal discretization can be unrolled into iterative neural architectures that generate the coefficients of the basis of the resulting neural operator. By leveraging numerical analysis, we establish convergence error estimates for these numerically derived neural operators. Notably, the network size depends polynomial on the approximation error in , improving the nested exponential dependence derived in the general neural operator approximation and aligning with the numerical observations. Finally, numerical experiments demonstrate reduced errors and improved parameter efficiency.
cs.LG
Neural operators enable fast PDE forecasting, but repeated predictions accumulate errors and fine-scale structures remain difficult to resolve. We introduce a frequency-decomposed finite-time flow-map neural operator (FNO) that leverages updated low-frequency features to guide nonlinear refinement of high-frequency information. Within each layer, this cross-scale conditioning connects global spectral processing with local detail refinement. The model directly predicts states at specified future times and adjusts the contributions of the two branches according to the prediction interval. For longer trajectories, it combines parallel predictions within short temporal segments with recursive propagation between segments. Experiments on five PDE benchmarks demonstrate improved forecasting accuracy over autoregressive and direct-prediction baselines. Ablations show that frequency-decomposed refinement can improve accuracy with fewer parameters, while the benefits of segmentation depend on spatial resolution and dynamical regime.