Deep learning of free boundary and Stefan problems

TL;DR

This paper introduces a physics-informed neural network framework to accurately solve and invert Stefan free boundary problems involving moving interfaces, demonstrating high effectiveness across various benchmarks.

Contribution

It develops a multi-network deep learning approach for forward and inverse Stefan problems, enabling accurate approximation of solutions and free boundaries from sparse data.

Findings

Successfully solves complex Stefan problems with moving boundaries

Reconstructs solutions and boundaries from noisy measurements

Achieves high accuracy in benchmark tests

Abstract

Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/DeepStefan}.