Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation

TL;DR

This work tackles the challenge of solving high-dimensional incompressible Navier–Stokes, Cahn–Hilliard, and their coupled CHNS equations using forward–backward stochastic neural networks (FBSNNs). By recasting these PDEs as FBSDEs and applying boundary-aware schemes (Dirichlet, Neumann, and periodic) alongside a stabilized, diagonalizable CH formulation, the authors train neural nets to approximate the solution and its gradient across the domain. Key contributions include boundary-reflection techniques for Neumann conditions, a parabolic reformulation of CH enabling decoupled FBSDEs, and a CHNS solver based on diagonalization into separate forward processes. Numerical experiments demonstrate stability and accuracy in high dimensions (up to 100D CH) and on classical NS benchmarks, with training times on standard hardware. Overall, the approach provides a scalable, unsupervised framework for solving complex, high-dimensional PDEs in fluid dynamics and multi‑phase flows, potentially enabling applications beyond conventional grid-based methods.

Abstract

Efficient algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the curse of dimensionality. We extend the forward-backward stochastic neural networks (FBSNNs) which depends on forward-backward stochastic differential equation (FBSDE) to solve incompressible Navier-Stokes equation. For Cahn-Hilliard equation, we derive a modified Cahn-Hilliard equation from a widely used stabilized scheme for original Cahn-Hilliard equation. This equation can be written as a continuous parabolic system, where FBSDE can be applied and the unknown solution is approximated by neural network. Also our method is successfully developed to Cahn-Hilliard-Navier-Stokes (CHNS) equation. The accuracy and stability of our methods are shown in many numerical experiments, specially in high dimension.

Figures (14)

• Figure 1: Illustration of Algorithm \ref{['alg:Algorithm 1']} for solving the incompressible Navier-Stokes equation
• Figure 2: Illustration of Algorithm \ref{['alg:Algorithm 2']} for solving the Cahn-Hilliard equation
• Figure 3: Network structure for periodic boundary condition
• Figure 4: Relative $L^{2}$ errors of $\mathbf{u}$ and training losses of the Algorithm \ref{['alg:Algorithm 1']} for \ref{['eq:Taysolu1']} with different $\nu$.
• Figure 5: Relative $L^{2}$ errors and training losses of the Algorithm \ref{['alg:Algorithm 1']} for \ref{['eq:ABCflow']} with different $\nu$.

Theorems & Definitions (1)

• Remark 1