Spectral-inspired Operator Learning with Limited Data and Unknown Physics

• URL: http://arxiv.org/abs/2505.21573v2
• Date: Fri, 26 Sep 2025 14:09:49 GMT
• Title: Spectral-inspired Operator Learning with Limited Data and Unknown Physics
• Authors: Han Wan, Rui Zhang, Hao Sun,
• Abstract summary: Spectral-Inspired Neural Operator (SINO) can model complex systems from just 2-5 trajectories without requiring explicit PDE terms.<n>To model nonlinear effects, SINO employs a Pi-block that performs multiplicative operations on spectral features, complemented by a low-pass filter to suppress aliasing.<n>Experiments on both 2D and 3D PDE benchmarks demonstrate that SINO achieves state-of-the-art performance, with improvements of 1-2 orders of magnitude in accuracy.
• Score: 10.143396024546368
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Learning PDE dynamics from limited data with unknown physics is challenging. Existing neural PDE solvers either require large datasets or rely on known physics (e.g., PDE residuals or handcrafted stencils), leading to limited applicability. To address these challenges, we propose Spectral-Inspired Neural Operator (SINO), which can model complex systems from just 2-5 trajectories, without requiring explicit PDE terms. Specifically, SINO automatically captures both local and global spatial derivatives from frequency indices, enabling a compact representation of the underlying differential operators in physics-agnostic regimes. To model nonlinear effects, it employs a Pi-block that performs multiplicative operations on spectral features, complemented by a low-pass filter to suppress aliasing. Extensive experiments on both 2D and 3D PDE benchmarks demonstrate that SINO achieves state-of-the-art performance, with improvements of 1-2 orders of magnitude in accuracy. Particularly, with only 5 training trajectories, SINO outperforms data-driven methods trained on 1000 trajectories and remains predictive on challenging out-of-distribution cases where other methods fail.

Related papers

• Out-of-distribution generalization of deep-learning surrogates for 2D PDE-generated dynamics in the small-data regime [1.9116784879310027]
  We study autoregressive deep-learning surrogates for two-dimensional PDE dynamics on periodic domains.<n>In small-data periodic 2D PDE settings, convolutional architectures with inductive biases aligned to locality remain strong contenders for accurate and moderately out-of-distribution-robust surrogate modeling.
  arXiv  Detail & Related papers  (2026-01-13T10:20:59Z)
• Stable spectral neural operator for learning stiff PDE systems from limited data [12.62991453201434]
  We introduce an equation-free learning framework, namely, the Stable Spectral Neural Operator (SSNO)<n>SSNO embeds spectrally inspired structures in its architecture, yielding strong inductive biases for learning the underlying physics.<n>It achieves prediction errors one to two orders of magnitude lower than leading models.
  arXiv  Detail & Related papers  (2025-12-12T16:09:38Z)
• Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators [0.0]
  We propose Neural Graph Simulators (GNS) as a principled surrogate modeling paradigm for time-dependent partial differential equations (PDEs)<n>GNS leverages message-passing combined with numerical time-stepping schemes to learn PDE dynamics by modeling the instantaneous time derivatives.<n>Results demonstrate that GNS is markedly more data-efficient, achieving less than 1% relative L2 error using only 3% of available trajectories.
  arXiv  Detail & Related papers  (2025-09-07T17:54:23Z)
• SING: SDE Inference via Natural Gradients [0.0]
  We propose SDE Inference via Natural Gradients (SING) to efficiently exploit the underlying geometry of the model and variational posterior.<n>SING enables fast and reliable inference in latent SDE models by approximating intractable integrals and parallelizing computations in time.<n>We show that SING outperforms prior methods in state inference and drift estimation on a variety of datasets.
  arXiv  Detail & Related papers  (2025-06-21T19:36:11Z)
• Enabling Automatic Differentiation with Mollified Graph Neural Operators [73.52999622724101]
  We propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries.<n>For a PDE example on regular grids, $m$GNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences.<n>It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough.
  arXiv  Detail & Related papers  (2025-04-11T06:16:30Z)
• Mechanistic PDE Networks for Discovery of Governing Equations [52.492158106791365]
  We present Mechanistic PDE Networks, a model for discovery of partial differential equations from data.<n>The represented PDEs are then solved and decoded for specific tasks.<n>We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations.
  arXiv  Detail & Related papers  (2025-02-25T17:21:44Z)
• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [60.58067866537143]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis.<n>To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers.<n> Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• Characteristic Performance Study on Solving Oscillator ODEs via Soft-constrained Physics-informed Neural Network with Small Data [6.3295494018089435]
  This paper compares physics-informed neural network (PINN), conventional neural network (NN) and traditional numerical discretization methods on solving differential equations (DEs) We focus on the soft-constrained PINN approach and formalized its mathematical framework and computational flow for solving Ordinary DEs and Partial DEs. We demonstrate that the DeepXDE-based implementation of PINN is not only light code and efficient in training, but also flexible across CPU/GPU platforms.
  arXiv  Detail & Related papers  (2024-08-19T13:02:06Z)
• Physics-informed Discretization-independent Deep Compositional Operator Network [1.2430809884830318]
  We introduce a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters and irregular domain shapes. Inspired by deep operator neural networks, our model involves a discretization-independent learning of parameter embedding repeatedly. Numerical results demonstrate the accuracy and efficiency of the proposed method.
  arXiv  Detail & Related papers  (2024-04-21T12:41:30Z)
• DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training [87.90342423839876]
  We present a new auto-regressive denoising pre-training strategy, which allows for more stable and efficient pre-training on PDE data. We train our PDE foundation model with up to 0.5B parameters on 10+ PDE datasets with more than 100k trajectories.
  arXiv  Detail & Related papers  (2024-03-06T08:38:34Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Spectral operator learning for parametric PDEs without data reliance [6.7083321695379885]
  We introduce a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The proposed framework demonstrates superior performance compared to existing scientific machine learning techniques.
  arXiv  Detail & Related papers  (2023-10-03T12:37:15Z)
• Physics-constrained robust learning of open-form partial differential equations from limited and noisy data [1.50528618730365]
  This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data.
  arXiv  Detail & Related papers  (2023-09-14T12:34:42Z)
• Geometry-Informed Neural Operator for Large-Scale 3D PDEs [76.06115572844882]
  We propose the geometry-informed neural operator (GINO) to learn the solution operator of large-scale partial differential equations. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points.
  arXiv  Detail & Related papers  (2023-09-01T16:59:21Z)
• Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics [97.38308257547186]
  Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and material models. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. We introduce a new framework termed "Neural Constitutive Laws" (NCLaw) which utilizes a network architecture that strictly guarantees standard priors.
  arXiv  Detail & Related papers  (2023-04-27T17:42:24Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)
• Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs [70.51212431290611]
  Partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. We propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Phi41 model and the 2d Navier-Stokes equation.
  arXiv  Detail & Related papers  (2022-04-13T08:53:41Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.