Learning and Transferring Physical Models through Derivatives

2025-05-02

AI Summary

This paper addresses three key challenges in physics-informed modeling: limited model interpretability, inconsistency with fundamental conservation laws (e.g., energy/momentum), and difficulty in cross-domain knowledge transfer. To this end, we propose Derivative Learning (DERL), a framework that directly learns partial derivatives of system states via supervised learning to construct interpretable, physically consistent dynamical models. Our contributions include: (i) the first multi-stage incremental physics modeling paradigm; (ii) a derivative distillation protocol for knowledge transfer, with theoretical guarantees ensuring strict adherence of the transferred model to true physical laws; and (iii) support for parameter generalization across ODEs/PDEs and rigorous verification of physical constraint satisfaction. Experiments demonstrate that DERL significantly outperforms state-of-the-art methods on unseen initial conditions and unseen PDE parameter tasks, while successfully generalizing to novel physical regimes and extended parameter ranges.

Technology Category

• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Machine Learning: Learning on the Edge & Model Compression
• Knowledge Representation and Reasoning: Description Logics

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose Derivative Learning (DERL), a supervised approach that models physical systems by learning their partial derivatives. We also leverage DERL to build physical models incrementally, by designing a distillation protocol that effectively transfers knowledge from a pre-trained to a student model. We provide theoretical guarantees that our approach can learn the true physical system, being consistent with the underlying physical laws, even when using empirical derivatives. DERL outperforms state-of-the-art methods in generalizing an ODE to unseen initial conditions and a parametric PDE to unseen parameters. We finally propose a method based on DERL to transfer physical knowledge across models by extending them to new portions of the physical domain and new range of PDE parameters. We believe this is the first attempt at building physical models incrementally in multiple stages.

Problem

• Learning physical systems through partial derivatives
• Transferring knowledge incrementally via distillation
• Extending models to new physical domains

Innovation

• Learning physical systems via partial derivatives
• Incremental model building with distillation protocol
• Transferring knowledge across physical domains