[论文解读] GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media

Matthaios Chatzopoulos, Phaedon-Stelios Koutsourelakis|arXiv (Cornell University)|Feb 16, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

GenPANIS 是一个统一的潜变量生成框架，将离散微观结构嵌入连续潜在空间，以解决多相介质中的正向和反 PDE 问题，实现基于梯度的推断并通过高效的物理感知解码器进行原则性不确定性量化。

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.

研究动机与目标

• Motivate and address the challenge of learning with discrete, high-dimensional microstructures in multiphase media.
• Develop a unified generative model that supports both forward prediction and inverse inference within a single architecture.
• Preserve discrete microstructures while enabling gradient-based inference through continuous latent embeddings.
• Leverage unlabeled data, virtual data (PDE residuals), and limited labeled data to train a physics-consistent surrogate.
• Provide uncertainty quantification and extrapolation capabilities under varying boundary conditions and microstructures.

提出的方法

• Embed discrete microstructures into a continuous latent space z to enable differentiable inference without relaxing the microstructure.
• Use a learnable prior p_theta(z) via a RealNVP normalizing flow to capture multimodal latent structure.
• Implement decoders p_theta(x|z) for discrete microstructures (logistic PCA for binary; categorical for multi-phase) and p_theta(u|z) for PDE solutions through a physics-aware coarse-grained decoder.
• Couple z to PDE inputs via a neural mapping f_theta but map to PDE outputs through a coarse solution operator Y(X) and lifting h(Y) to obtain y, yielding a differentiable physics-informed bottleneck.
• Train with three data modalities—unlabeled microstructures, virtual PDE residual data, and labeled microstructure–PDE solution pairs—via a single ELBO-based objective (SVI).
• Enable forward and inverse tasks within the same model by conditioning in latent space and performing posterior inference over z (e.g., via HMC) to generate x or predict u.]

实验结果

研究问题

• RQ1 Can a latent-variable GenPANIS framework recover discrete microstructures from sparse or noisy PDE observations (inverse problem) without sacrificing physical fidelity?
• RQ2 Does a unified generative approach with a physics-aware decoder outperform task-specific models for forward and inverse PDE problems in multiphase media?
• RQ3 How well does the model extrapolate to unseen boundary conditions, volume fractions, and microstructure morphologies, with limited labeled data?
• RQ4 Can the framework provide principled uncertainty quantification for both forward predictions and inverse recoveries in multiphase systems?

主要发现

• GenPANIS maintains accuracy in extrapolative scenarios including unseen boundary conditions, volume fractions, and microstructures.
• The method outperforms state-of-the-art approaches while using 10–100 times fewer parameters.
• It provides principled uncertainty quantification through posterior sampling in the latent space.
• Training leverages unlabeled data, virtual data (PDE residuals), and limited labeled data, reducing reliance on large labeled datasets.
• A physics-aware, differentiable coarse-grained PDE decoder preserves governing equations and supports extrapolation to varying conditions.
• Posterior inference in the latent space enables gradient-based inference for inverse problems and data-efficient forward predictions.