Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers

2025-05-20

AI Summary

Existing reduced-order models (ROMs) for high-dimensional time-varying PDE systems suffer from poor generalizability and prediction drift due to inconsistencies between learned latent dynamics and discrete physical constraints. Method: This paper proposes a Physics-Embedded Differentiable ROM, integrating a differentiable PDE solver (implemented in JAX) as a hard constraint within the latent neural dynamics training pipeline—thereby enforcing adherence to discretized physical laws during learning. The approach jointly combines parameterized manifold learning, physics-informed loss design, and data assimilation under sparse observations. Results: Evaluated on multiple PDE benchmarks, the method substantially outperforms state-of-the-art data-driven and physics-informed ROMs: it enables robust cross-parameter extrapolation, long-term stable predictions beyond training time horizons, accurate modeling with minimal training data, and high-fidelity field reconstruction from sparse, irregularly sampled observations.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Reduced-order modeling (ROM) of time-dependent and parameterized differential equations aims to accelerate the simulation of complex high-dimensional systems by learning a compact latent manifold representation that captures the characteristics of the solution fields and their time-dependent dynamics. Although high-fidelity numerical solvers generate the training datasets, they have thus far been excluded from the training process, causing the learned latent dynamics to drift away from the discretized governing physics. This mismatch often limits generalization and forecasting capabilities. In this work, we propose Physics-informed ROM ($Phi$-ROM) by incorporating differentiable PDE solvers into the training procedure. Specifically, the latent space dynamics and its dependence on PDE parameters are shaped directly by the governing physics encoded in the solver, ensuring a strong correspondence between the full and reduced systems. Our model outperforms state-of-the-art data-driven ROMs and other physics-informed strategies by accurately generalizing to new dynamics arising from unseen parameters, enabling long-term forecasting beyond the training horizon, maintaining continuity in both time and space, and reducing the data cost. Furthermore, $Phi$-ROM learns to recover and forecast the solution fields even when trained or evaluated with sparse and irregular observations of the fields, providing a flexible framework for field reconstruction and data assimilation. We demonstrate the framework's robustness across different PDE solvers and highlight its broad applicability by providing an open-source JAX implementation readily extensible to other PDE systems and differentiable solvers.

Problem

• Accelerate simulation of high-dimensional systems via compact latent manifold learning
• Align latent dynamics with governing physics to enhance generalization and forecasting
• Enable robust field reconstruction from sparse, irregular observations using differentiable solvers

Innovation

• Incorporates differentiable PDE solvers into training
• Ensures latent dynamics align with governing physics
• Enables robust forecasting with sparse observations