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

• URL: http://arxiv.org/abs/2512.21633v1
• Date: Thu, 25 Dec 2025 11:27:40 GMT
• Title: MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations
• Authors: Qiuqi Li, Yiting Liu, Jin Zhao, Wencan Zhu,
• Abstract summary: Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems.<n>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.<n>We propose a novel and scalable framework that significantly enhances the Neural Galerkin Method (NGM) by incorporating the Meta-Auto-Decoder (MAD) paradigm.
• Score: 5.767740428776141
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Structure and asymptotic preserving deep neural surrogates for uncertainty quantification in multiscale kinetic equations [5.181697052513637]
  High dimensionality of kinetic equations with parameters poses computational challenges for uncertainty quantification (UQ)<n>Traditional Monte Carlo (MC) sampling methods suffer from slow convergence and high variance, which become increasingly severe as the dimensionality of the space grows.<n>We introduce surrogate models based on structure and preserving neural networks (SAPNNs)<n>SAPNNs are specifically designed to satisfy key physical properties, including positivity, conservation laws, entropy dissipation, parameter limits.
  arXiv  Detail & Related papers  (2025-06-12T12:20:53Z)
• KO: Kinetics-inspired Neural Optimizer with PDE Simulation Approaches [45.173398806932376]
  This paper introduces KO, a novel neural gradient inspired by kinetic theory and partial differential equation (PDE) simulations.<n>We reimagine the dynamics of network parameters as the evolution of a particle system governed by kinetic principles.<n>This physics-driven approach inherently promotes parameter diversity during optimization, mitigating the phenomenon of parameter condensation.
  arXiv  Detail & Related papers  (2025-05-20T18:00:01Z)
• Implicit Neural Differential Model for Spatiotemporal Dynamics [5.1854032131971195]
  We introduce Im-PiNDiff, a novel implicit physics-integrated neural differentiable solver for stabletemporal dynamics.<n>Inspired by deep equilibrium models, Im-PiNDiff advances the state using implicit fixed-point layers, enabling robust long-term simulation.<n>Im-PiNDiff achieves superior predictive performance, enhanced numerical stability, and substantial reductions in memory and cost.
  arXiv  Detail & Related papers  (2025-04-03T04:07:18Z)
• Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems [49.819436680336786]
  We propose an efficient transformed Gaussian process state-space model (ETGPSSM) for scalable and flexible modeling of high-dimensional, non-stationary dynamical systems.<n>Specifically, our ETGPSSM integrates a single shared GP with input-dependent normalizing flows, yielding an expressive implicit process prior that captures complex, non-stationary transition dynamics.<n>Our ETGPSSM outperforms existing GPSSMs and neural network-based SSMs in terms of computational efficiency and accuracy.
  arXiv  Detail & Related papers  (2025-03-24T03:19:45Z)
• MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation [48.41289705783405]
  We propose a PDE-embedded network with multiscale time stepping (MultiPDENet)<n>In particular, we design a convolutional filter based on the structure of finite difference with a small number of parameters to optimize.<n>A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction.
  arXiv  Detail & Related papers  (2025-01-27T12:15:51Z)
• A Stochastic Approach to Bi-Level Optimization for Hyperparameter Optimization and Meta Learning [74.80956524812714]
  We tackle the general differentiable meta learning problem that is ubiquitous in modern deep learning. These problems are often formalized as Bi-Level optimizations (BLO) We introduce a novel perspective by turning a given BLO problem into a ii optimization, where the inner loss function becomes a smooth distribution, and the outer loss becomes an expected loss over the inner distribution.
  arXiv  Detail & Related papers  (2024-10-14T12:10:06Z)
• Ensemble Kalman Filtering Meets Gaussian Process SSM for Non-Mean-Field and Online Inference [47.460898983429374]
  We introduce an ensemble Kalman filter (EnKF) into the non-mean-field (NMF) variational inference framework to approximate the posterior distribution of the latent states. This novel marriage between EnKF and GPSSM not only eliminates the need for extensive parameterization in learning variational distributions, but also enables an interpretable, closed-form approximation of the evidence lower bound (ELBO) We demonstrate that the resulting EnKF-aided online algorithm embodies a principled objective function by ensuring data-fitting accuracy while incorporating model regularizations to mitigate overfitting.
  arXiv  Detail & Related papers  (2023-12-10T15:22:30Z)
• Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning [0.0]
  We present a physics-informed machine-learning (PIML) approach for the approximation of slow invariant manifold (SIMs) of singularly perturbed systems. We show that the proposed PIML scheme provides approximations, of equivalent or even higher accuracy, than those provided by other traditional GSPT-based methods. A comparison of the computational costs between symbolic, automatic and numerical approximation of the required derivatives in the learning process is also provided.
  arXiv  Detail & Related papers  (2023-09-14T14:10:22Z)
• Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks [0.0]
  This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs. We propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme. We show that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios.
  arXiv  Detail & Related papers  (2023-08-03T08:14:28Z)
• AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios [51.94807626839365]
  We propose the attention-inspired numerical solver (AttNS) to solve differential equations due to limited data.<n>AttNS is inspired by the effectiveness of attention modules in Residual Neural Networks (ResNet) in enhancing model generalization and robustness.
  arXiv  Detail & Related papers  (2023-02-05T01:39:21Z)
• Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions [0.0]
  This work presents a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification. The proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations.
  arXiv  Detail & Related papers  (2022-11-13T01:57:18Z)
• Neural Stochastic Dual Dynamic Programming [99.80617899593526]
  We introduce a trainable neural model that learns to map problem instances to a piece-wise linear value function. $nu$-SDDP can significantly reduce problem solving cost without sacrificing solution quality.
  arXiv  Detail & Related papers  (2021-12-01T22:55:23Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)

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.