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

Zijie Li, Saurabh Patil, Francis Ogoke, Dule Shu, Wilson Zhen, Michael Schneier, John R. Buchanan,, Amir Barati Farimani

TL;DR

The paper introduces Latent Neural PDE Solver (LNS), a reduced-order framework that learns time-dependent PDE dynamics in a coarse latent space. It first trains a nonlinear autoencoder to project full-order fields to a coarse latent grid and then learns a propagator to forecast latent states autoregressively, reducing training cost and enabling longer backpropagation horizons. Across 2D Navier–Stokes, shallow water, and tank sloshing problems, LNS achieves competitive accuracy with significantly lower compute than full-order neural PDE surrogates, and performance improves with longer training rollouts, especially for challenging cases. The approach highlights the practicality of latent-space dynamics for scalable neural PDE surrogates and points to future extensions to arbitrary meshes and geometries.

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.

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

TL;DR

The paper introduces Latent Neural PDE Solver (LNS), a reduced-order framework that learns time-dependent PDE dynamics in a coarse latent space. It first trains a nonlinear autoencoder to project full-order fields to a coarse latent grid and then learns a propagator to forecast latent states autoregressively, reducing training cost and enabling longer backpropagation horizons. Across 2D Navier–Stokes, shallow water, and tank sloshing problems, LNS achieves competitive accuracy with significantly lower compute than full-order neural PDE surrogates, and performance improves with longer training rollouts, especially for challenging cases. The approach highlights the practicality of latent-space dynamics for scalable neural PDE surrogates and points to future extensions to arbitrary meshes and geometries.

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.
Paper Structure (34 sections, 21 equations, 14 figures, 7 tables, 2 algorithms)

This paper contains 34 sections, 21 equations, 14 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Neural PDE solver
  4. Two-stage model for image compression and synthesis
  5. Latent space modelling for time-dependent PDE problems
  6. Methodology
  7. Problem definition
  8. Autoencoder for discretized field's dimensionality reduction
  9. Baseline neural PDE solver
  10. Fourier neural operator
  11. UNet
  12. Conditioning
  13. Conditioning for a feed-forward network
  14. Conditioning for a spectral convolution layer
  15. Conditioning for a convolution block
  16. ...and 19 more sections

Figures (14)

  • Figure 1: (a) An autoencoder is trained to project the input field to latent field with much coarser discretization. (b) A neural network is trained to predict the latent field at different time steps autoregressively.
  • Figure 2: Description of the tank sloshing system
  • Figure 3: Tank Sloshing: Sample prediction of $x$ component of the velocity field $\mathbf{u}$ on varying height dataset with liquid surface height: $h=25\%$. Unit: $m/s$.
  • Figure 4: Tank Sloshing: Sample prediction $x$ component of the velocity field $\mathbf{u}$ on varying frequency dataset with oscillation frequency: $\omega=7.12 ~\text{rad/s}$. Unit: $m/s$.
  • Figure 5: Shallow Water: Comparison on models' predicted $y-$component of $\mathbf{u}$ with $5$ training rollout steps. UNet's prediction exhibits notable artifacts after certain time steps. Unit: $m/s$.
  • ...and 9 more figures