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Physics-informed reduced order model with conditional neural fields

Minji Kim, Tianshu Wen, Kookjin Lee, Youngsoo Choi

TL;DR

This work introduces a physics-informed, reduced-order modeling framework (CNF-ROM) for parametrized PDEs by coupling a parametric neural ODE (PNODE) that evolves a latent state with a coordinate-based decoder conditioned on this state. It advances exact initial and boundary condition enforcement via extended R-function approximate distance functions and an auxiliary derivative network to stabilize higher-order derivatives, enabling simultaneous learning of latent dynamics and the decoder. The approach supports interpolation, extrapolation in parameter space, and temporal forecasting, demonstrated on the 1D Burgers equation with improved performance when applying PINN fine-tuning for unseen parameters. Overall, CNF-ROM provides a robust, geometry-aware, ROM approach that incorporates both data and physics while ensuring exact IC/BC satisfaction, with potential applicability to higher-dimensional problems.

Abstract

This study presents the conditional neural fields for reduced-order modeling (CNF-ROM) framework to approximate solutions of parametrized partial differential equations (PDEs). The approach combines a parametric neural ODE (PNODE) for modeling latent dynamics over time with a decoder that reconstructs PDE solutions from the corresponding latent states. We introduce a physics-informed learning objective for CNF-ROM, which includes two key components. First, the framework uses coordinate-based neural networks to calculate and minimize PDE residuals by computing spatial derivatives via automatic differentiation and applying the chain rule for time derivatives. Second, exact initial and boundary conditions (IC/BC) are imposed using approximate distance functions (ADFs) [Sukumar and Srivastava, CMAME, 2022]. However, ADFs introduce a trade-off as their second- or higher-order derivatives become unstable at the joining points of boundaries. To address this, we introduce an auxiliary network inspired by [Gladstone et al., NeurIPS ML4PS workshop, 2022]. Our method is validated through parameter extrapolation and interpolation, temporal extrapolation, and comparisons with analytical solutions.

Physics-informed reduced order model with conditional neural fields

TL;DR

This work introduces a physics-informed, reduced-order modeling framework (CNF-ROM) for parametrized PDEs by coupling a parametric neural ODE (PNODE) that evolves a latent state with a coordinate-based decoder conditioned on this state. It advances exact initial and boundary condition enforcement via extended R-function approximate distance functions and an auxiliary derivative network to stabilize higher-order derivatives, enabling simultaneous learning of latent dynamics and the decoder. The approach supports interpolation, extrapolation in parameter space, and temporal forecasting, demonstrated on the 1D Burgers equation with improved performance when applying PINN fine-tuning for unseen parameters. Overall, CNF-ROM provides a robust, geometry-aware, ROM approach that incorporates both data and physics while ensuring exact IC/BC satisfaction, with potential applicability to higher-dimensional problems.

Abstract

This study presents the conditional neural fields for reduced-order modeling (CNF-ROM) framework to approximate solutions of parametrized partial differential equations (PDEs). The approach combines a parametric neural ODE (PNODE) for modeling latent dynamics over time with a decoder that reconstructs PDE solutions from the corresponding latent states. We introduce a physics-informed learning objective for CNF-ROM, which includes two key components. First, the framework uses coordinate-based neural networks to calculate and minimize PDE residuals by computing spatial derivatives via automatic differentiation and applying the chain rule for time derivatives. Second, exact initial and boundary conditions (IC/BC) are imposed using approximate distance functions (ADFs) [Sukumar and Srivastava, CMAME, 2022]. However, ADFs introduce a trade-off as their second- or higher-order derivatives become unstable at the joining points of boundaries. To address this, we introduce an auxiliary network inspired by [Gladstone et al., NeurIPS ML4PS workshop, 2022]. Our method is validated through parameter extrapolation and interpolation, temporal extrapolation, and comparisons with analytical solutions.

Paper Structure

This paper contains 10 sections, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Models: Parameterized CNF-ROMs
  3. Conditional neural fields
  4. Decoder
  5. PNODE
  6. Methods: Physics-informed training with exact satisfaction of IC/BC
  7. Physics-informed learning with parametrized CNF-ROM
  8. Exact imposition of IC and BC
  9. Training objectives
  10. Numerical results and discussion

Figures (2)

  • Figure 1: Diagram of the CNF-ROM structure, combining PNODE with the architecture of yin2023continuous.
  • Figure 2: Performance verification through: (top left) loss trajectories, (bottom left) evaluation on training and test set parameters, and temporal extrapolation at $t=1.25$; (top right) heatmap of loss for model (a) at $\boldsymbol{\mu} = 20$; (bottom right) heatmap of loss for model (b) at $\boldsymbol{\mu} = 20$.