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
physics.geo-ph
Sea level fingerprints connect the spatial distribution of rapid ice mass change to sea surface height (SSH) change, but the spatial resolving power of this inverse problem remains poorly quantified. Here we provide the first systematic, scale-resolved quantification of Greenland Ice Sheet mass change structure recoverable from its SSH fingerprint when ice loss geometry varies. We develop a physics-informed neural operator, constrained by an accurate neural-operator emulator of the forward sea level equation, to probe the information content of this ill-conditioned inverse problem. Across synthetic ice change ensembles, broad ice loss patterns can be recovered, but distinct ice fields can produce nearly identical fingerprints, so fine-scale structure depends on the assumed a-priori ice change distribution. Inversion fidelity varies with the wavelength and amplitude of ice mass change. An empirical error law separates these effects and defines a scale-dependent accuracy ceiling. For a satellite-altimetry-derived reference field, the estimated fingerprint-only ceiling is under the present SSH observation geometry and matched-prior assumption. The leading three principal components explain 99.76% of fingerprint variance, yet higher-order components retain information needed to distinguish finer ice change structure. SSH observations near Greenland carry most information on ice loss locations. Applying the leading fingerprint principal components to the observed 1993–2019 SSH rate recovers the large-scale Greenland fingerprint, while the residual field is correlated with basin-scale bathymetry, consistent with unresolved sterodynamic variability. These results establish a quantitative empirical bound on fingerprint-only ice-loss resolving power and identify how independent observations and ice sheet constraints can improve inference.
eess.SY
We address the problem of improving the performance of pre-stabilized networked systems subject to plug-and-play (PnP) operations, namely additions or removals of subsystems. While distributed baseline controllers can preserve stability after a PnP transition, augmenting them with expressive neural policies can invalidate the formal stability guarantees. To overcome this issue, we build on recently proposed Performance Boosting controllers based on the Internal Model Control (IMC) principle, which reduce closed-loop stability to a finite -gain requirement on an auxiliary operator. In the distributed setting, this operator is formed by interconnecting local neural operators that exchange information only between neighboring subsystems. We then derive free parametrizations of this interconnection, namely explicit maps from unconstrained trainable variables to interconnected neural operators satisfying a prescribed network gain bound for every parameter value. The obtained distributed neural controller can therefore be trained by unconstrained gradient-based optimization without stability constraints or projection steps, while retaining the prescribed gain bound at every training iterate. We develop two complementary parametrizations. A neighbor-based compositional parametrization uses local coupling bounds, so that after a PnP event only subsystems whose bounds change need to update their gain parameters. A network-norm parametrization instead uses a bound on the overall coupling matrix and can reduce conservatism, at the price of requiring network-wide information. A PnP DC-microgrid case study shows that the proposed neural layer improves voltage transients under topology changes and load variations while preserving nominal closed-loop stability.
cs.LG
The Long-Range Ising Model benchmark (LRIM) guarantees long-range dependence by construction: each label is the node's spin times the spin field convolved with a power-law coupling, so any model's distance from the target is measurable. We prove that a readout whose filter is a polynomial in the adjacency, the linear core of isotropic message passing and spectral graph networks, leaves an error floor that no degree lowers, because the target varies along a level set of the spectrum on which every such filter is constant. Computed exactly, the floor and the best filter of each degree predict two spectral networks adapted from published architectures to within a tenth of a decade. A filter escapes the floor only if it weighs separately the offsets the adjacency lumps together, and must still resolve the coupling across the whole lattice. A strided filter of about fifty coefficients does both: fitted to the training labels, its test error is 52 times below the best published network's. A dilated convolutional network beats the best published result at 256 nodes, matches it at 1,024, and gives the first in-domain results at the three larger sizes, while a Fourier neural operator with its standard modes falls further behind as the lattice grows. At its standard budget the dilated network stays two to four orders of magnitude short of a larger strided filter fitted to the labels. The analysis carries over to the benchmark's second coupling.
cs.LG
Physics-informed neural networks (PINNs) reduce or eliminate the need for large labelled datasets by enforcing governing equations during training. Physics-informed Deep Operator Networks (PI-DeepONets) extend this approach to learning solution operators, enabling rapid prediction across families of problems. Despite these advantages, we show that their application to higher-dimensional problems remains limited by the computational cost of training. To address this issue, we propose a modified PI-DeepONet architecture that reduces this cost by removing redundancy in how the Branch network processes input functions across collocation points. The proposed formulation is mathematically equivalent to the original architecture but reduces input tensor sizes and the number of Branch network evaluations. We analyse the computational complexity of both architectures and derive theoretical scaling laws for training time with respect to the number of collocation points and input functions. The proposed architecture is evaluated on 1D, 2D, and 3D linear advection problems. In all cases, it achieves comparable predictive accuracy to the original PI-DeepONet while consistently reducing training time. In 2D experiments, a maximum speed-up of is observed, with empirical training-time scaling agreeing with the predicted linear dependence on collocation points and input functions. Furthermore, training a 3D model on an NVIDIA A100 GPU is only feasible using the proposed architecture, as the original formulation exceeds memory limits. These results demonstrate that careful consideration of how PI-DeepONets are implemented can yield substantial gains, enhancing the practical performance of PI-DeepONets and enabling their application to otherwise intractable physics-informed learning problems.
cs.AI
Neural operators approximate solution operators of partial differential equations at a fraction of the cost of numerical solvers. Efficient grid-based architectures such as the Fourier neural operator (FNO) assume uniform rectangular grids, whereas engineering simulations use large, irregular, and locally refined meshes. We introduce L-SplineFNO, a memory-efficient neural operator that couples an FNO backbone with a sparse, learnable tensor-product B-spline transfer between unstructured meshes and a latent Cartesian grid. A local normalised scatter avoids the geometry-dependent solve of analytic spline transformation while preserving compact support, partition of unity, and spline smoothness by construction. The neural operator, conditioned on global design and operating parameters, learns the mapping of geometry to the field solution. Across six three-dimensional engineering benchmarks, L-SplineFNO achieves strong predictive accuracy while requiring substantially less memory and compute than baseline operators, scaling to 26.6 million nodes in training and 46 million in inference on a single A100 GPU. It also supports zero-shot super-resolution and efficient closed-form spatial derivatives. Uniquely, the analytic spline decoder yields rigorous prediction bounds over continuous space and uncertain parameters, via Bernstein convex hulls and Taylor models, demonstrated on a 29.2-million-node MAST-U tokamak model.
physics.comp-ph
Morphodynamic simulations require repeated coupling between hydrodynamic solvers, sediment transport modules, and bed updating. For long duration, multi-scenario simulations on large unstructured meshes, such coupling is computationally demanding. This study proposes an Adaptive-Horizon Graph Neural Operator (AH-GNO) that formulates bed evolution as a graph operator learning problem from recent hydrodynamic and topographic states to predict future bed elevation increments. The model operates directly on the native unstructured meshes. The approach integrates area-weighted graph learning, historical state selection, and forecast horizon estimation, allowing the effective history length (K) and update interval (B) to vary with the evolving flow and bed state. Using Telemac2D/GAIA solutions as supervised numerical references, the framework is evaluated on the official Bump and Yen bend benchmarks and on a field-scale application (Waiho River, New Zealand). Results show that the AH-GNO reproduces the main patterns of bedform migration, bend-scale erosion and deposition, and localized bed adjustment. Across the three cases, the tested CPU-GPU deployment reduced post-training bed-update workflow time from 12~1204 s with GAIA to 0.08~3.34 s with AH-GNO. This improvement corresponds to deployment-specific accelerations of approximately 150~360 times relative to 4-core GAIA CPU configurations, and with 105~186 times gains for 20-core CPU configurations. These results suggest that, while retaining hydrodynamic feedback from TELEMAC-2D, a learned operator for bed updating can provide an efficient acceleration component for morphodynamic scenario analysis and rapid coupled simulation.
cs.LG
Neural PDE solvers commonly generate multi-step predictions through autoregressive rollout, recursively applying a learned evolution operator and feeding each prediction back as input for the next step. Approximation errors can therefore propagate through the rollout, causing predictions at a shared target time to depend on their temporal origins and rollout depths. For a well-posed deterministic PDE flow, however, states at different times along the same physical trajectory should, under exact evolution, reach the same target state. To this end, we propose Align Latent Rollout (ALR), a regularization approach that promotes consistency across rollout origins through the neural PDE solver's internal representations. A second-order Taylor analysis of latent discrepancy propagation motivates two objectives: zeroth-order latent state consistency, which aligns representations at a shared target time, and first-order latent dynamics consistency, which aligns local evolution responses to shared perturbations. ALR can be integrated into different neural PDE solvers by regularizing suitable internal representations, without modifying their backbone architectures. Experiments across multiple PDE benchmarks and neural-operator backbones show that ALR improves prediction accuracy while reducing shared-target discrepancies across rollout origins.
cs.LG
Accurate medium- to long-range epidemic forecasting requires extrapolating high-dimensional, heterogeneous and nonstationary dynamics from partial observations. Yet hybrid models often leave implicit which observations inform which epidemic processes and how learned graph interactions modify system state. We introduce LE-EpiGNODE, which replaces generic message passing with process-specific, mechanism-guided neural operators: age-contact information drives Lagrangian infection generation, whereas mobility drives conservative Eulerian population transport within a structure-preserving universal differential equation. We evaluated LE-EpiGNODE across 14 controlled synthetic mechanisms spanning time-modulated, state-dependent, nonlinear and coupled dynamics, three complementary COVID-19 settings, 21 comparator models and five targeted ablations. Forecasting gains tended to become more pronounced at longer horizons; at the longest horizon, LE-EpiGNODE achieved the lowest WMAPE in all three real-world datasets, reducing error by 17.7-40.0% relative to the best baseline. Controlled ablations linked these gains to the Lagrangian-Eulerian decomposition and process-aligned priors, while synthetic ground truth showed that learned operators and induced state fluxes track known generating dynamics. These results support mechanism-organized graph neural dynamics as a route to reliable long-horizon extrapolation with directly testable, process-level mechanistic interpretation beyond trajectory accuracy.
cs.SD
Head-related transfer function (HRTF) spatial upsampling from sparse measurements is a fundamental challenge in personalized binaural audio. We formulate HRTF upsampling as an operator learning problem and introduce SPARO, a neural operator that maps sparse measurements at a set of spatial directions to the HRTF field, a continuous representation that can be queried at arbitrary positions. SPARO projects the measurements onto a learned set of spherical latent modes, processes the latent modes with Transformer blocks, and decodes the resulting representations at query directions. On SONICOM, using the LAP24 measurement configuration, SPARO attains the best log-spectral distortion (LSD) and interaural level difference (ILD) error at every sparsity level from 3 to 100 measurements (18% lower LSD and 43% lower ILD error than the strongest baseline at 100 measurements) while remaining competitive in interaural time difference (ITD) error. It reconstructs a dense HRTF 15 to 373 times faster than the leading neural-field method RANF. We further validate the approach on HUTUBS, demonstrating its effectiveness across HRTF datasets.
cs.LG
Neural partial differential equation (PDE) surrogates trained with mean squared error (MSE) can capture dominant dynamics but often lose high-frequency information and fine-scale structures. Generative models (e.g., Flow Matching) can capture the distribution of fine-scale details, yet stochastic errors may accumulate during autoregressive rollouts and compromise long-term stability. To address these limitations, we introduce Refine and Align(ReAl), a model-agnostic plug-in that augments pretrained PDE surrogates to improve the reconstruction of high-frequency structures associated with complex physical phenomena. In the refinement stage, a conditional flow-matching model learns the residual distribution around the deterministic base prediction, recovering missing fine-scale structures while retaining the dominant dynamics. In the alignment stage, to mitigate the generative discrepancies amplified in the autoregressive rollout, particularly in chaotic systems, we generate multiple trajectories from the same input and reinforce the refiner with trajectory-level rewards, thereby achieving long-horizon dynamical alignment. Experiments across diverse surrogate architectures and PDE tasks demonstrate that ReAl consistently improves high-frequency fidelity, forecasting accuracy, and long-horizon stability.
cs.LG
Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations. Despite remarkable architectural diversity, these methods often share a common input representation: raw coordinate-value pairs . We ask whether inputs aligned with PDE differential structure can improve learning. Through Picard iteration on mild solutions, we show that derivatives of the input appear explicitly in the solution operator, suggesting that providing derivative features should reduce the network's implicit differentiation burden. We prove this intuition: providing derivative features improves approximation bounds from to , where is parameter count, is input regularity, is the PDE order, and is spatial dimension. Our central finding, however, is a surprise: the optimal derivative order is strictly less than the PDE order . This gap arises from a bias-variance tradeoff in spectral space that we characterize in closed form. Learning from data achieves automatic spectral regularization. We introduce -NO (del-NO), for derivative-augmented neural operators, an architecture-agnostic augmentation that provides learnable fractional derivative features to any neural operator backbone. Across benchmark problems and architectures, -NO consistently improves prediction accuracy, with learned orders that reflect a representation of known physics modulated by noise and finite-sample constraints.
cs.LG
We present several advances on neural operators by viewing the action of operator layers as the minimizers of Bregman regularized optimization problems over Banach function spaces. The proposed framework allows interpreting the activation operators as Bregman proximity operators from dual to primal space. This novel viewpoint is general enough to recover classical neural operators as well as a new variant, coined Bregman neural operators, which includes the inverse activation operator and features the same expressivity of standard neural operators. Numerical experiments support the added benefits of the Bregman variant of Fourier neural operators for training deeper and more accurate models.
physics.comp-ph
Rapid prediction of pressure-driven flow in microfluidic porous media is important for experimental design, tracer-test interpretation, and digital-twin development for subsurface energy and critical mineral applications. High-fidelity simulations resolve geometry-controlled flow but are computationally expensive across large geometry ensembles. Here, we introduce GeONet, a Geometry-Encoded Operator Network that predicts steady-state pressure fields by jointly representing material properties, spatial coordinates, and solid–void geometry. GeONet extends the Branch–Trunk architecture with a dedicated Leaf Network encoding the binary domain mask through multiscale features. The Branch Network represents porosity and permeability, the Trunk Network combines coordinates with local geometric descriptors and random Fourier features, and a gated DeepMixer integrates these representations to predict pressure. GeONet was trained and evaluated using 5,000 PFLOTRAN simulations of heterogeneous Reduced Complexity Platform domains, comprising nearly 30 TB of data. Comparisons with pore-scale Stokes solutions across three geometries yielded Pearson correlations of 0.987–0.991 and values of 0.974–0.982, supporting PFLOTRAN as reference data. Under matched partitions and two-stage hyperparameter optimization, GeONet substantially outperformed a boundary-condition-aware DeepONet baseline. Across three held-out domains, GeONet achieved NRMSE of 0.0381–0.0500, MAPE of 1.76–2.59%, and of 0.9673–0.9822, versus 0.1483–0.2040, 7.81–10.45%, and 0.5658–0.7126 for DeepONet, respectively. On three additional unseen geometries, GeONet achieved NRMSE of 0.0448–0.0676 and of 0.9511–0.9770. These results demonstrate improved accuracy, pressure-range preservation, and cross-geometry generalization through explicit geometry encoding.
cs.IT
Digital-domain self-interference cancellation is essential in in-band full-duplex radios after passive suppression and analog cancellation. It is particularly challenging when the desired signal is strong relative to the residual self-interference. In this regime, conventional iterative digital cancellers update their coefficients from desired-signal-contaminated received samples and may suffer substantial performance degradation. This paper reformulates digital-domain self-interference cancellation as an operator-learning problem. The proposed framework learns the temporal evolution of the residual self-interference channel from historical observations and uses the learned operator together with the known local transmit signal to reconstruct the current residual self-interference. A Transformer neural operator is used to realize this framework. The resulting method is trained in a desired-signal-free setting and then applied in the presence of a desired signal. A continuity-based analysis characterizes the sensitivity of the learned operator to the resulting input perturbation. Numerical experiments in static and time-varying channels, including time-varying SIR conditions, demonstrate stable cancellation performance over a wide range of interference levels. At 30 dB SIR, the proposed method improves the cancellation performance by 5–15 dB over the considered conventional methods.
physics.comp-ph
We present a global-in-time spectral Duhamel-Picard method with polynomial expansions for post-exposure bake (PEB) simulation. The coupled acid-quencher reaction-diffusion equations are reformulated as nonlinear Volterra integral equations using the Duhamel principle. Full trajectories are represented by finite polynomial expansions, with temporal coefficient fields determined over the bake interval by Anderson-accelerated Picard iteration, thereby avoiding sequential time marching. In 3-D PEB simulations, we employ Chebyshev and Legendre temporal bases while evaluating the Duhamel integrals using the corresponding Gaussian quadrature rules. We demonstrate that the proposed method simulates a large domain of voxels on the order of one minute. The representation is further incorporated into a physics-informed neural operator (PINO) with a mixed-transform variant of a factorized Fourier neural operator, which is trained without precomputed solution trajectories. Together, the integral formulation and its PINO extension enable accelerated, time-marching-free prediction of concentration fields at arbitrary times within the bake interval.
cs.CE
Deep learning has enabled substantial advances in oncology, including cancer detection and diagnosis, medical image analysis, tumor characterization, treatment planning, response assessment, and outcome prediction. However, conventional data-driven models often require large annotated datasets, exhibit limited generalization across heterogeneous clinical settings, and lack explicit mechanisms for incorporating the physical, physiological, and biological principles underlying cancer development and treatment. Physics-informed deep learning (PIDL) integrates governing physical laws, mechanistic models, physiological constraints, and domain knowledge into deep learning frameworks, providing a principled approach for learning from sparse, noisy, and heterogeneous oncological data. This review presents a task-oriented synthesis of PIDL methods across oncology, encompassing neural networks, Kolmogorov-Arnold networks, convolutional and vision architectures, neural operators, generative models, and hybrid physics-data-driven frameworks. Their applications are examined across cancer detection and diagnosis, image reconstruction and analysis, tumor characterization, quantitative imaging, tumor progression, treatment response, radiation therapy, treatment planning, and multimodal clinical modeling. Based on the synthesized methodological and application-level evidence, we develop task-optimized PIDL workflows that align the clinical task with appropriate data representations, physical constraints, model architectures, and validation strategies. We further analyze key challenges, including physical-model fidelity, computational scalability, uncertainty quantification, domain generalization, standardized benchmarking, and clinical validation.
stat.ML
This paper studies Wong–Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong–Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.
cs.LG
This chapter provides a mathematical introduction to operator surrogates, which are finite-parametric maps approximating complex (typically nonlinear) operators between infinite-dimensional (function) spaces. A typical approach to constructing operator surrogates is the encoder-approximator-decoder paradigm, whereby infinite-dimensional inputs are mapped to a finite-dimensional latent space, processed, and then mapped back to the infinite-dimensional output space. We discuss various methods for creating efficient representation systems, including stable linear frameworks like Banach frames and data-driven techniques such as Principal Component Analysis (PCA), Active Subspaces, and nonlinear autoencoders. Subsequently, we survey key surrogate architectures, contrasting spectral methods, based on polynomial chaos expansions and related technologies, with more recent neural network approaches. We further discuss architectures that do not rely on representation systems, such as Fourier Neural Operators and transformers. We elaborate on their relationship and how they relate to other neural network operator surrogates, particularly those obtained by so-called "algorithm-unrolling". Additionally, we provide an introduction to the theoretical analysis of approximation and generalization errors. We explain how underlying low-dimensional structures, provided for example by holomorphy, are essential to overcome the so-called "curse of dimensionality": this refers to the fact that the complexity of numerical algorithms tends to increase exponentially in the problem dimensionality. We present convergence rate bounds for holomorphic operators that depend on the number of trainable parameters and the regularity of the input and output data. Throughout, we prioritize intuition over detailed technical proofs and provide precise pointers to the literature for detailed arguments.
physics.flu-dyn
High-fidelity three-dimensional turbulent flow fields are essential for scientific and engineering analyses, yet obtaining them through direct numerical simulation or high-resolution measurements remains expensive. Super-resolution reconstruction from coarse-grained data therefore offers a promising alternative, but remains fundamentally challenging because fine-scale structures are intrinsically three-dimensional and dynamically coupled across scales. Existing data-driven approaches still struggle to recover multiscale volumetric structures and their coherent temporal evolution. To address these limitations, we propose a turbulence-aware framework for time-resolved three-dimensional flow reconstruction. The framework couples a multiscale neural operator, designed to recover volumetric structures, with a state space model to promote temporal consistency in the reconstructed dynamics. In addition, Navier-Stokes residuals are incorporated as soft constraints to regularize the reconstruction toward physically plausible dynamics. A mixture-of-experts mechanism is also introduced to support unsupervised adaptation to unknown coarse-graining. Across evaluations on canonical turbulent systems and atmospheric reanalysis data, the proposed framework reduces reconstruction errors relative to learning-based and classical baselines, while better preserving multiscale spatial structure and coherent temporal evolution. These results validate the effectiveness of the proposed framework in reconstructing high-fidelity three-dimensional flow fields from coarse-grained data.
cs.LG
Neural operator splitting composes a library of neural operators, one per physical mechanism, into zero-shot simulators of new equations. Existing approaches, however, require these mechanisms and their labels to be specified in advance. We ask whether a pretrained library can instead be adapted and grown from unlabeled trajectories in which multiple mechanisms act simultaneously. We introduce SPLICE, a framework that discovers operators without mechanism labels on mixture data. The contributions of the candidate operators to each trajectory are inferred by solving a convex optimization problem. A calibrated test then assigns each discovered operator the mechanism it implements. Failure cases are also detectable, when the library lacks the required physics or when mechanisms are insufficiently separable. Across one and two dimensional systems, the discovered operators recover mechanisms from unlabeled mixtures and compose into zero-shot simulators of unseen equations; a frozen library can also grow an operator for physics absent from pretraining, without labels and without forgetting. On Navier-Stokes, the discovered operators compose about as accurately as their supervised counterparts.