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MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations

arXiv 2512.21633 ↗First posted 25 Dec 2025cs.LGcited by 0

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Abstract (arXiv)

Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods, often face challenges in generalization and long-time prediction efficiency due to their dependence on full space-time approximations. To address these issues, we propose a novel and scalable framework that significantly enhances the Neural Galerkin Method (NGM) by incorporating the Meta-Auto-Decoder (MAD) paradigm. Our approach leverages space-time decoupling to enable more stable and efficient time integration, while meta-learning-driven adaptation allows rapid generalization to unseen parameter configurations with minimal retraining. Furthermore, randomized sparse updates effectively reduce computational costs without compromising accuracy. Together, these advancements enable our method to achieve physically consistent, long-horizon predictions for complex parameterized evolution equations with significantly lower computational overhead. Numerical experiments on benchmark problems demonstrate that our methods performs comparatively well in terms of accuracy, robustness, and adaptability.

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References · 28

  1. 2017Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networkscited by 4,861
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16 more
  1. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  2. 2017Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanicscited by 213
  3. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
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  5. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  6. 2020Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningcited by 77
  7. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  8. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 55
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  11. 2022SVD Perspectives for Augmenting DeepONet Flexibility and Interpretabilitycited by 35
  12. 2023Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networkscited by 18
  13. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  14. 2022VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equationscited by 1
  15. 2025LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian processcited by 1
  16. 2025Filtered Neural Galerkin model reduction schemes for efficient propagation of initial condition uncertainties in digital twinscited by 1

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