Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media

Matthaios Chatzopoulos, Phaedon-Stelios Koutsourelakis

TL;DR

GenPANIS is a unified generative framework that preserves exact discrete microstructures while enabling gradient-based inference through continuous latent embeddings, and outperforms state-of-the-art methods while using 10 - 100 times fewer parameters and providing principled uncertainty quantification.

Abstract

Inverse problems and inverse design in multiphase media, i.e., recovering or engineering microstructures to achieve target macroscopic responses, require operating on discrete-valued material fields, rendering the problem non-differentiable and incompatible with gradient-based methods. Existing approaches either relax to continuous approximations, compromising physical fidelity, or employ separate heavyweight models for forward and inverse tasks. We propose GenPANIS, a unified generative framework that preserves exact discrete microstructures while enabling gradient-based inference through continuous latent embeddings. The model learns a joint distribution over microstructures and PDE solutions, supporting bidirectional inference (forward prediction and inverse recovery) within a single architecture. The generative formulation enables training with unlabeled data, physics residuals, and minimal labeled pairs. A physics-aware decoder incorporating a differentiable coarse-grained PDE solver preserves governing equation structure, enabling extrapolation to varying boundary conditions and microstructural statistics. A learnable normalizing flow prior captures complex posterior structure for inverse problems. Demonstrated on Darcy flow and Helmholtz equations, GenPANIS maintains accuracy on challenging extrapolative scenarios - including unseen boundary conditions, volume fractions, and microstructural morphologies, with sparse, noisy observations. It outperforms state-of-the-art methods while using 10 - 100 times fewer parameters and providing principled uncertainty quantification.

GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media

TL;DR

GenPANIS is a unified generative framework that preserves exact discrete microstructures while enabling gradient-based inference through continuous latent embeddings, and outperforms state-of-the-art methods while using 10 - 100 times fewer parameters and providing principled uncertainty quantification.

Abstract

Inverse problems and inverse design in multiphase media, i.e., recovering or engineering microstructures to achieve target macroscopic responses, require operating on discrete-valued material fields, rendering the problem non-differentiable and incompatible with gradient-based methods. Existing approaches either relax to continuous approximations, compromising physical fidelity, or employ separate heavyweight models for forward and inverse tasks. We propose GenPANIS, a unified generative framework that preserves exact discrete microstructures while enabling gradient-based inference through continuous latent embeddings. The model learns a joint distribution over microstructures and PDE solutions, supporting bidirectional inference (forward prediction and inverse recovery) within a single architecture. The generative formulation enables training with unlabeled data, physics residuals, and minimal labeled pairs. A physics-aware decoder incorporating a differentiable coarse-grained PDE solver preserves governing equation structure, enabling extrapolation to varying boundary conditions and microstructural statistics. A learnable normalizing flow prior captures complex posterior structure for inverse problems. Demonstrated on Darcy flow and Helmholtz equations, GenPANIS maintains accuracy on challenging extrapolative scenarios - including unseen boundary conditions, volume fractions, and microstructural morphologies, with sparse, noisy observations. It outperforms state-of-the-art methods while using 10 - 100 times fewer parameters and providing principled uncertainty quantification.
Paper Structure (44 sections, 44 equations, 19 figures, 11 tables, 2 algorithms)

This paper contains 44 sections, 44 equations, 19 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Generative Model Architecture
  4. Learnable prior via normalizing flows.
  5. Decoder for the PDE-input $p_{\boldsymbol{\theta}}(\boldsymbol{x} \mid \boldsymbol{z})$
  6. The decoder for the PDE-output $p_{\boldsymbol{\theta}}(u | \boldsymbol{z})$
  7. Data types and Model Likelihoods
  8. Unlabeled data
  9. Virtual data
  10. Labeled data
  11. Training with Stochastic Variational Inference (SVI)
  12. Predictive Estimates for Forward and Inverse Problems
  13. Inverse Problems
  14. Forward Problems
  15. Evaluation Metrics
  16. ...and 29 more sections

Figures (19)

  • Figure 1: Probabilistic graphical model illustration for the GenPANIS framework. Plates indicate different data types (labeled, unlabeled, virtual). See text for details on model structure and parameterization.
  • Figure 2: Forward problem predictions of the three models.
  • Figure 3: Reference posterior statistics for Darcy flow obtained from one million HMC samples. Full observations were used with $SNR=100$.
  • Figure 4: Solving the Darcy flow inverse problem for various levels of noise by observing the full solution field. No additional residual information is used after training the models.
  • Figure 5: Solving the Helmholtz equation inverse problem for various levels of noise by observing the full solution field. No additional residual information is used after training the models.
  • ...and 14 more figures