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We propose a level-set approach to characterize the region occupied by the solid in Stefan problems with and without surface tension, based on their recent probabilistic reformulation. The level-set function is parameterized by a feed-forward neural network, whose parameters are trained using the probabilistic formulation of the Stefan growth condition. The algorithm can handle Stefan problems where the liquid is supercooled and can capture surface tension effects through the simulation of particles along the moving boundary together with an efficient approximation of the mean curvature. We demonstrate the effectiveness of the method on a variety of examples with and without radial symmetry.
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This content will become publicly available on March 5, 2026
Dual Neural Network (DuNN) method for elliptic partial differential equations and systems
This paper presents the Dual Neural Network (DuNN) method, a physics-driven numerical method designed to solve elliptic partial differential equations and systems using deep neural network functions and a dual formulation. The underlying elliptic problem is formulated as an optimization of the complementary energy functional in terms of the dual variable, where the Dirichlet boundary condition is weakly enforced in the formulation. To accurately evaluate the complementary energy functional, we employ a novel discrete divergence operator. This discrete operator preserves the underlying physics and naturally enforces the Neumann boundary condition without penalization. For problems without reaction term, we propose an outer-inner iterative procedure that gradually enforces the equilibrium equation through a pseudo-time approach.
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- Award ID(s):
- 2110571
- PAR ID:
- 10610926
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Computational and Applied Mathematics
- Volume:
- 467
- Issue:
- C
- ISSN:
- 0377-0427
- Page Range / eLocation ID:
- 116596
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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