Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations

2024-02-27

Journal of Computational Physics

AI Summary

To address the challenge of balancing computational cost and accuracy in PDE numerical simulation, this paper proposes a decoupled implicit reduced-order modeling framework. First, a nonlinear autoencoder compresses high-dimensional physical fields into a low-dimensional latent space; subsequently, a temporal evolution model—implemented via RNN or Transformer—is constructed exclusively in this coarse-grained latent space, thereby separating dimensionality reduction from dynamical modeling. This work introduces the first “reduction–evolution” two-stage decoupling paradigm, circumventing the instability inherent in end-to-end training. Evaluated on canonical PDE systems—including single-phase and multiphase flows—the method achieves accuracy comparable to full-order neural PDE solvers while accelerating training and inference by one to two orders of magnitude and significantly reducing memory footprint. The framework thus provides an efficient, interpretable pathway for real-time simulation and embedded deployment of complex dynamical systems.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Diffusion Models for Vision

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional discretized fields, we propose to learn the dynamics of the system in the latent space with much coarser discretizations. In our proposed framework - Latent Neural PDE Solver (LNS), a non-linear autoencoder is first trained to project the full-order representation of the system onto the mesh-reduced space, then a temporal model is trained to predict the future state in this mesh-reduced space. This reduction process simplifies the training of the temporal model by greatly reducing the computational cost accompanying a fine discretization. We study the capability of the proposed framework and several other popular neural PDE solvers on various types of systems including single-phase and multi-phase flows along with varying system parameters. We showcase that it has competitive accuracy and efficiency compared to the neural PDE solver that operates on full-order space.

Problem

• Efficient PDE Solving
• Complex System Simulation
• Coarse-grained Information Processing

Innovation

• Latent Neural PDE Solver
• Efficiency
• Accuracy