A shallow physics-informed neural network for solving partial differential equations on surfaces

TL;DR

This work develops a shallow physics-informed neural network (PINN) to solve PDEs on surfaces by embedding surface differential operators in Cartesian coordinates, enabled by a level-set representation of the geometry to compute normals and mean curvature. The stationary case solves Laplace–Beltrami and diffusion-type equations on static surfaces using a single hidden layer and a Loss_{\Delta_s} residual, which yields high accuracy (down to ~10^{-6} relative $L^2$ error) across complex geometries. The time-dependent and evolving-surface cases extend the framework with continuous-time and discrete-time (Runge–Kutta) schemes, plus a surface-network that learns a homeomorphism from $\mathbb{S}^2$ to the evolving surface, enabling joint tracking of geometry and solution. Applications include surfactant transport on droplets under shear and surface heating, with results demonstrating robust, mesh-free performance and physically plausible predictions even under large deformations. The approach offers a compact, easy-to-train alternative to traditional mesh-based or embedding methods for surface PDEs with potential impact in fluid-structure interactions and interface dynamics.

Abstract

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We perform a series of performance study for the present methodology by solving Laplace-Beltrami equation and surface diffusion equation on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. Then we extend the present methodology to solve the advection-diffusion equation on an evolving surface with given velocity. To track the surface, we additionally introduce a prescribed hidden layer to enforce the topological structure of the surface and use the network to learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate the surfactant transport on the droplet surface under shear flow and obtain some physically plausible results.

Figures (7)

• Figure 1: Shapes of ellipsoid and torus (top row), genus-2 torus and cheese-like (bottom row). The color code denotes the magnitude of mean curvature $H$.
• Figure 2: Time history of training $\hbox{Loss}_{\Delta_s}$ with $N = 40$ using different optimizers. Dash-dotted line: ADAM, dashed line: L-BFGS, solid line: LM. All training processes use $M = 400$ training points.
• Figure 3: Prediction solution $u_\mathcal{N}$ and corresponding absolute error $|u_\mathcal{N}-u|$ with $N = 60$ neurons employed. From left to right: torus, genus-2 surface, cheese-like surface.
• Figure 4: The snapshots of solution distribution for heating up the cheese-like surface at different times. The color code ranging from 0 to 0.36 indicates the magnitude of the solution.
• Figure 5: The snapshots of the network solution $\mathbf{x}_\mathcal{N}$ with $N = 40$ neurons at different times. The color code indicates the magnitude of the mean curvature $H_\mathcal{N}$.