FBNO Deep Neural Operator for Free Boundary Problems

Free boundary problems (FBPs), 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 significant limitations. While recent advances in neural operators have revolutionized PDE solving by learning mappings between function spaces, existing frameworks remain constrained to predefined domains, rendering them inapplicable to FBPs. Here, we introduce the Free Boundary Neural Operator (FBNO), a universal framework that overcomes this fundamental limitation by leveraging topological conjugacy between dynamical systems. FBNO approximates both the flow map of a conjugate system and the homeomorphism linking it to the original FBP, enabling predictions on evolving domains without prior geometric knowledge. Crucially, we provide an approximation theorem guaranteeing the method’s theoretical feasibility. The efficacy of FBNO has been comprehensively demonstrated in numerical experiments spanning phase transitions, non-convex geometries, and multi-physics systems. The work marks a turning point in free boundary simulations, where neural networks unlock both speed and precision.