Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete geometric elements (usually triangles). Recently, physics-informed neural networks (PINNs) have emerged as a continuous, mesh-free alternative that does not suffer from FEM's sensitivity to mesh quality or geometric discretization errors. We present PINNSur, a simple framework for using PINNs on curved surfaces: we train a neural field to approximate the surface's normals, and then we express surface differential operators using their projection from $\mathbb{R}^3$ onto the surface. Since every orientable manifold has well-defined normals, our method is suitable for all such surfaces, regardless of curvature or topology, enabling many geometry processing applications. Moreover, despite their empirical success in solving PDEs in flat Euclidean domains, PINNs lack convergence guarantees to the true solution of the underlying PDE, and there is limited systematic experimental evidence demonstrating such convergence. This gap restricts their adoption as reliable solvers compared to established methods like FEM, where convergence to the true solution is well understood and theoretically grounded. These surface PDEs are particularly challenging to solve convergently, as one must not only deal with the convergence of the function approximation, but also with the convergence of the geometric approximation of the surface itself. In this work, we empirically investigate the convergence behavior of PINNs for solving surface PDEs by introducing a simple empirical convergence test.

翻译：曲面上的偏微分方程是科学计算与几何处理中的基础问题。求解曲面偏微分方程的一种常见方法是有限元法，该方法将曲面离散为若干几何单元（通常为三角形）。近年来，物理信息神经网络作为一种连续、无网格的替代方案崭露头角，它避免了有限元法对网格质量敏感或存在几何离散误差的问题。我们提出PINNSur——一个用于在曲面上应用物理信息神经网络的简洁框架：训练一个神经场来近似曲面的法向量，然后通过将曲面微分算子从$\mathbb{R}^3$投影至曲面来实现其表达。由于每个可定向流形均具有定义明确的法向量，我们的方法适用于所有此类曲面（无论曲率或拓扑结构如何），从而支持多种几何处理应用。然而，尽管物理信息神经网络在平坦欧几里得域中求解偏微分方程取得了实证成功，但其对底层偏微分方程真实解的收敛性缺乏理论保证，且缺乏系统性实验证据来论证这种收敛性。这一缺陷限制了其作为可靠求解器的应用——相比之下，有限元法等成熟方法对真实解的收敛性具有明确的理论基础与完善的理解。因此，解法式的曲面偏微分方程尤其具有挑战性：不仅需要处理函数逼近的收敛性，还需要处理曲面本身几何逼近的收敛性。本文通过引入一个简单的经验收敛性测试，实证研究了物理信息神经网络在求解曲面偏微分方程时的收敛行为。