mlsanityData to 7 Oct 2026

Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers

arXiv 2606.02623 ↗First posted 29 May 2026cs.NE cs.AI cs.LGcited by 0

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Abstract (arXiv)

Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.

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References · 32

  1. 2023Mamba: Linear-Time Sequence Modeling with Selective State Spacescited by 4,616
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2021Efficiently Modeling Long Sequences with Structured State Spacescited by 2,424
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  5. 2022Simplified State Space Layers for Sequence Modelingcited by 776
  6. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  7. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  8. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  9. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
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  11. 2020Neural Controlled Differential Equations for Irregular Time Seriescited by 501
  12. 2023Resurrecting Recurrent Neural Networks for Long Sequencescited by 397
20 more
  1. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  2. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 233
  3. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
  4. 2020Neural Rough Differential Equations for Long Time Seriescited by 140
  5. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  6. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 122
  7. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  8. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 109
  9. 2022Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Samplingcited by 71
  10. 2022Failure-informed adaptive sampling for PINNscited by 64
  11. 2023PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networkscited by 62
  12. 2021Evolutional Deep Neural Networkcited by 48
  13. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  14. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 41
  15. 2024Oscillatory State-Space Modelscited by 35
  16. 2024Log Neural Controlled Differential Equations: The Lie Brackets Make a Differencecited by 34
  17. 2023Physics-informed neural networks for solving forward and inverse problems in complex beam systemscited by 24
  18. 2022Random Weight Factorization Improves the Training of Continuous Neural Representationscited by 13
  19. 2025Sub-Sequential Physics-Informed Learning with State Space Modelcited by 10
  20. 2025Neuro-Spectral Architectures for Causal Physics-Informed Networkscited by 3

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  1. 2023LatentPINNs: Generative physics-informed neural networks via a latent representation learning
  2. 2025Merging Memory and Space: A State Space Neural Operator
  3. 2025KD-PINN: Knowledge-Distilled PINNs for ultra-low-latency real-time neural PDE solvers
  4. 2025Towards a Foundation Model for Partial Differential Equations Across Physics Domains
  5. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers
  6. 2026Autoregression-Free Neural Operators for Time-Dependent PDEs
  7. 2024Domain-decoupled Physics-informed Neural Networks with Closed-form Gradients for Fast Model Learning of Dynamical Systems
  8. 2024On the Benefits of Memory for Modeling Time-Dependent PDEs