Disentangled Representation Learning for Parametric Partial Differential Equations

Area of Science:

• Machine Learning
• Scientific Computing
• Physics-Informed AI

Background:

• Neural operators (NOs) are powerful tools for approximating solutions to partial differential equations (PDEs).
• However, their black-box nature limits interpretability, hindering the understanding of underlying physical mechanisms.
• Extracting physical parameters from NOs remains a challenge for inverse problem-solving.

Purpose of the Study:

• To develop a method for learning disentangled representations from neural operator parameters.
• To enable neural operators to effectively solve inverse problems.
• To enhance the physical interpretability and generalization capabilities of neural operator frameworks.

Main Methods:

• Introduction of DisentangO, a novel hyper-neural operator architecture.
• Utilizing a multi-task neural operator to distill varying PDE parameters via a task-wise adaptive layer.
• Employing a variational autoencoder to disentangle these variations into identifiable latent factors.

Main Results:

• DisentangO successfully unveils and disentangles latent physical factors from neural operator parameters.
• The method demonstrates effective extraction of meaningful and interpretable latent features across supervised, semi-supervised, and unsupervised learning.
• Empirical evaluations confirm enhanced physical interpretability and robust generalization across diverse systems.

Conclusions:

• DisentangO bridges the gap between predictive performance and physical understanding in neural operator frameworks.
• The proposed approach offers a new paradigm for solving inverse problems using neural operators.
• This work significantly advances the interpretability and applicability of neural operators in scientific discovery.