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

2026-02-16

AI Summary

This work addresses the challenge of jointly modeling forward and inverse problems for partial differential equations (PDEs) in multiphase media, where discrete microstructures induce non-differentiability and conventional approaches suffer from either low physical fidelity or model redundancy. To this end, we propose GenPANIS, a unified generative framework that enables end-to-end optimization by embedding discrete microstructures into continuous latent variables, thereby facilitating gradient backpropagation while strictly preserving structural discreteness. Integrating a physics-aware decoder, a learnable normalizing flow prior, and a differentiable coarse-grained PDE solver, GenPANIS achieves unsupervised training, extrapolation generalization, and uncertainty quantification with minimal labeled data. In Darcy flow and Helmholtz equation tasks, GenPANIS outperforms existing methods with 10–100× fewer parameters, maintaining high accuracy and reliable uncertainty estimates under unseen boundary conditions, volume fractions, and microstructural morphologies.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• inverse problems
• multiphase media
• discrete-valued material fields
• non-differentiable optimization
• microstructure design

Innovation

• generative modeling
• inverse problems
• discrete microstructures
• physics-informed neural networks
• normalizing flows