FROST: LEARNING FREE-BOUNDARY EQUILIBRIA FOR SIMULATION AND INVERSE DESIGN

Free-boundary problems are partial differential equations (PDEs) whose domain evolves and is not known in advance; they govern solidification, growth and transport across the physical sciences. Classical sharp-interface solvers remesh as the boundary moves, lose robustness when domains merge or split, and are recomputed from scratch for every new instance, while existing neural operators act on a fixed, known domain or track the boundary by a single smooth deformation and so cannot represent changes in topology. We introduce FROST, a free-boundary equilibrium operator that represents the solution as a joint fixed point of the physical field and a level set on one fixed background grid, in which the physical free-boundary condition is the fixed-point residual. Because the geometry is a level set rather than a deformation, a single operator represents merging, splitting and nucleation without remeshing. In its equilibrium realisation the fixed point is solved and trained by implicit differentiation in the manner of a deep equilibrium model, so training memory is constant in the number of solver steps and convergence is monitored; a conditioned realisation of the same operator predicts time-dependent and parametric families in a single pass. The trained operator is then differentiated with respect to a design variable, so forward simulation and inverse design reduce to one problem and the physics residual serves as a trust signal for the design. Across steady and time-dependent free-boundary benchmarks FROST recovers boundaries and the changes in topology that a single deformation cannot represent, learns from only a handful of simulations, and steers a merging front to a prescribed target.