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Hyperbolic mean curvature flow computed by physics-informed neural networks

Shuangshuang Duan, Chunlei He, Shoujun Huang, Dexing Kong

TL;DR

This work addresses the numerical solution of hyperbolic mean curvature flow (HMCF) for evolving plane curves and closed surfaces using a mesh-free physics-informed neural network (PINN) framework. By formulating residuals for curves and surfaces that encode the normal hyperbolic evolution and energy terms, and by enforcing topology through soft constraints, the authors develop a two-pronged training strategy: a multi-stage Adam/L-BFGS scheme for curves with curriculum learning, and a two-phase Adam with OneCycleLR followed by L-BFGS for surfaces. Numerical experiments across circles, ellipses, spheres, ellipsoids, and tori, with constant and non-constant initial velocities, demonstrate high-precision relative $L^2$ errors (often in the $10^{-4}$ to $10^{-3}$ range) and illustrate how the dissipative parameter $\beta$ mediates a hyperbolic-to-parabolic transition. The results establish PINNs as a viable, mesh-free tool for hyperbolic geometric flows and motivate further exploration of PINN-based solvers for high-dimensional PDEs with complex topology.

Abstract

In this paper, we explore the evolution of plane curves and surfaces governed by the hyperbolic mean curvature flow. We propose a mesh-free approach based on the physics-informed neural networks (PINNs), which eliminates the need for discretization and meshing of computational domains, and is efficient in solving partial differential equations involving high dimensions. To the best of our knowledge, this is the first result on the numerical analysis by employing the PINNs for the hyperbolic geometric evolution equations in the literature. The effectiveness of this method is demonstrated through several numerical simulations by selecting diverse initial curves and surfaces, as well as both constant and non-constant initial velocities.

Hyperbolic mean curvature flow computed by physics-informed neural networks

TL;DR

This work addresses the numerical solution of hyperbolic mean curvature flow (HMCF) for evolving plane curves and closed surfaces using a mesh-free physics-informed neural network (PINN) framework. By formulating residuals for curves and surfaces that encode the normal hyperbolic evolution and energy terms, and by enforcing topology through soft constraints, the authors develop a two-pronged training strategy: a multi-stage Adam/L-BFGS scheme for curves with curriculum learning, and a two-phase Adam with OneCycleLR followed by L-BFGS for surfaces. Numerical experiments across circles, ellipses, spheres, ellipsoids, and tori, with constant and non-constant initial velocities, demonstrate high-precision relative errors (often in the to range) and illustrate how the dissipative parameter mediates a hyperbolic-to-parabolic transition. The results establish PINNs as a viable, mesh-free tool for hyperbolic geometric flows and motivate further exploration of PINN-based solvers for high-dimensional PDEs with complex topology.

Abstract

In this paper, we explore the evolution of plane curves and surfaces governed by the hyperbolic mean curvature flow. We propose a mesh-free approach based on the physics-informed neural networks (PINNs), which eliminates the need for discretization and meshing of computational domains, and is efficient in solving partial differential equations involving high dimensions. To the best of our knowledge, this is the first result on the numerical analysis by employing the PINNs for the hyperbolic geometric evolution equations in the literature. The effectiveness of this method is demonstrated through several numerical simulations by selecting diverse initial curves and surfaces, as well as both constant and non-constant initial velocities.
Paper Structure (9 sections, 4 theorems, 59 equations, 19 figures, 4 tables, 2 algorithms)

This paper contains 9 sections, 4 theorems, 59 equations, 19 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Properties enjoyed by the flow
  3. Hyperbolic mean curvature flows for plane curves
  4. Hyperbolic mean curvature flow for surfaces
  5. The Algorithm of PINNs
  6. Numerical experiments
  7. Hyperbolic mean curvature flow for curves
  8. Hyperbolic mean curvature flow for surfaces
  9. Conclusions

Key Result

Proposition 2.1

Assume the initial velocity is normal. Then, the hyperbolic mean curvature flow (eq12) is normal.

Figures (19)

  • Figure 1: Comparison of the PINNs and analytical solution of the HMCF starting from a unit circle. Examples with $r_1 = 0$ on the left, and $r_1 = 1$ on the right. The relative $\mathbb{L}_2$ errors are 1.91e-4 on the left, and 3.68e-4 on the right, respectively. The size of the deep neural network is $7\times50$.
  • Figure 2: Comparisons of the PINNs and analytical solution of the HMCF starting from a unit circle. We show the evolution of radius over time for the initial velocity $r_1 = 1$, above the deep neural network size is $1 \times 50$, $3 \times 50$, $5 \times 50$ and $7 \times 50$ (from left to right), blew the deep neural network size is $1 \times 100$, $3 \times 100$, $5 \times 100$ and $7 \times 100$ (from left to right).
  • Figure 3: Loss functions for the HMCF with (a) $r_1=0$ and (b) $r_1=1$. The deep neural network size is $7\times50$.
  • Figure 4: HMCF starting from an ellipse. We show the evolution for $r_1=0$, $r_1=1$, and $r_1=-1$ (from left to right).
  • Figure 5: HMCF starting from an ellipse. We show the evolution over time for $r_1=sinu$ (left) and $r_1=cosu$(right).
  • ...and 14 more figures

Theorems & Definitions (7)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2