Deep neural operator for free boundary problems

Abstract Free boundary problems, characterized by partial differential equations defined on a priori unknown domains, arise across diverse scientific and engineering disciplines, from quantum physics to biomedical applications. Traditional numerical approaches face crucial limitations. Although recent advances in neural operators have revolutionized the solving of partial differential equations by learning mappings between function spaces, existing frameworks remain constrained to predefined domains, rendering them inapplicable to free boundary problems. Here we introduce the free boundary neural operator, a universal framework that overcomes this fundamental limitation by leveraging topological conjugacy between dynamical systems. The free boundary neural operator approximates both the flow map of a conjugate system and the homeomorphism linking it to the original free boundary problem, which enables predictions on evolving domains without prior geometric knowledge. Crucially, we provide an approximation theorem guaranteeing the theoretical feasibility of the method. We have comprehensively demonstrated the efficacy of the free boundary neural operator in numerical experiments spanning phase transitions, non-convex geometries and multi-physics systems. The work marks a turning point in free boundary simulations, as the use of neural networks unlocks both speed and precision.