Physics-Constrained Learning for Scientific Discovery: Inference in Differential Equations and Inverse Design via Generative Models

Abstract

Scientific discovery increasingly relies on machine learning (ML), but many scientific problems involve sparse data, physical constraints, and large combinatorial search spaces. In these regimes, ML can suffer from generalization and robustness issues, including generating outputs that violate physics constraints known a priori. Physics-informed machine learning aims to integrate physical constraints into ML frameworks with the goal of guaranteeing physically sound outputs, especially to scientific problems. In supervised machine learning, physics-informed neural networks (PINNs) have been influential in handling problems that traditional differential equation solvers struggle with; in model discovery, symbolic regression algorithms can discover closed-form models that best describe a dataset; within generative models, generative flow networks (GFlowNets) are used to perform inverse design of drugs, antibiotics, and materials.

This thesis develops physics-informed machine learning methods that integrate prior scientific knowledge into predictive, inverse, and generative models. We make three main contributions to physics-informed machine learning: a contribution in the domain of PINNs; a contribution in the domain of symbolic regression; a contribution to materials discovery via GFlowNets. In the domain of PINNs, we introduce Universal PINNs, which can be used to learn unknown components of differential equations from sparse or noisy data, and we apply them to discover the best form for the drug action of a chemotherapeutic, testing the method both on synthetic and experimental data (Chapters 2 and 3). Then, we integrate PINNs with conformal prediction, enabling PINNs to output confidence intervals with provable guarantees on both parameter fits and differential equation solutions (Chapter 4). For our contribution in the domain of symbolic regression (Chapter 5), we augment the efficiency of symbolic regression algorithms with dimensional analysis, and showcase the improvement in the performance of a well-known symbolic regression algorithm, PySR. In the domain of materials discovery (Chapter 6), we build a framework for catalyst discovery, for the application of hydrogen energy storage. In this work, we integrate ML-based relaxation, reward shaping, and action space constraints to generate stable and efficient catalysts. For two separate chemical reactions, we rediscover the best known catalysts within a constrained search space.

Taken together, these contributions show that physical constraints can improve sample efficiency and reliability in scientific machine learning. Future work will integrate Universal PINNs and uncertainty quantification more tightly; GFlowNets and differential equation models could be applied together to a new scientific problem; the catalyst discovery framework could be tested experimentally and augmented with an active learning loop, in order to discover truly new materials.