Reproduce “Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equations” on OpenResearch

About this paper

Yesom Park, Stanley Osher

Researchers at UCLA developed a neural implicit solution formula for Hamilton-Jacobi equations, which is learned by a deep neural network without pre-computed data. This method efficiently overcomes the curse of dimensionality, maintaining nearly constant computational resources for problems up to 40 dimensions, while accurately approximating viscosity solutions across convex, nonconvex, and state-dependent Hamiltonians.

Key insights

• The derived implicit solution formula is theoretically proven to coincide with the classical Hopf-Lax formula for convex Hamiltonians and initial conditions, ensuring it yields viscosity solutions in these cases.
• By leveraging implicit neural representations, the method effectively overcomes the curse of dimensionality, as computational time and memory scale primarily with network size rather than spatial discretization.
• The unsupervised training approach intrinsically incorporates initial conditions within the implicit formula's loss, eliminating the need for separate initial/boundary condition loss terms and their associated, often sensitive, regularization parameters.

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