Ultra Fast PDE Solving via Physics Guided Few-step Diffusion

2026-02-03

AI Summary

This work addresses the challenges of low sampling efficiency and insufficient physical consistency in existing diffusion models for solving partial differential equations (PDEs). The authors propose Phys-Instruct, a novel framework that uniquely integrates distribution matching with PDE-guided distillation. By leveraging a physics-informed distillation strategy, the method compresses a pre-trained diffusion solver into a highly efficient few-step generator while explicitly embedding differentiable PDE constraints to enforce physical fidelity. The resulting unconditional student model is both computationally efficient and readily adaptable to downstream tasks. Evaluated across five PDE benchmarks, Phys-Instruct achieves speedups of several orders of magnitude in inference time and reduces PDE errors by more than eightfold, substantially outperforming current diffusion-based baselines.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning on the Edge & Model Compression

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Diffusion-based models have demonstrated impressive accuracy and generalization in solving partial differential equations (PDEs). However, they still face significant limitations, such as high sampling costs and insufficient physical consistency, stemming from their many-step iterative sampling mechanism and lack of explicit physics constraints. To address these issues, we propose Phys-Instruct, a novel physics-guided distillation framework which not only (1) compresses a pre-trained diffusion PDE solver into a few-step generator via matching generator and prior diffusion distributions to enable rapid sampling, but also (2) enhances the physics consistency by explicitly injecting PDE knowledge through a PDE distillation guidance. Physic-Instruct is built upon a solid theoretical foundation, leading to a practical physics-constrained training objective that admits tractable gradients. Across five PDE benchmarks, Phys-Instruct achieves orders-of-magnitude faster inference while reducing PDE error by more than 8 times compared to state-of-the-art diffusion baselines. Moreover, the resulting unconditional student model functions as a compact prior, enabling efficient and physically consistent inference for various downstream conditional tasks. Our results indicate that Phys-Instruct is a novel, effective, and efficient framework for ultra-fast PDE solving powered by deep generative models.

Problem

• PDE solving
• diffusion models
• sampling cost
• physics consistency
• generative models

Innovation

• physics-guided diffusion
• few-step generation
• PDE distillation
• generative modeling
• physics consistency