We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like other walk on spheres (WoS) algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces, etc. ), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters---hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics.

中文翻译：

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球体上的差分行走


我们引入了一种蒙特卡洛方法，用于计算偏微分方程 （PDE） 解相对于问题参数（例如域几何或边界条件）的导数。可以在任意点计算导数，而无需执行全局求解或构造体积网格或网格。因此，该方法非常适合复杂几何的逆问题，例如 PDE 约束的形状优化。与其他球体上行走 （WoS） 算法一样，我们的方法很容易并行化，并且与边界表示（网格、样条、隐式曲面等）无关，支持较大的拓扑变化。我们特别关注筛选的泊松方程，这些方程对科学和几何计算中的各种问题进行建模。与可微渲染一样，我们联合估计所有参数的导数---因此，成本不会随着参数数量的增加而显著增加。在实践中，即使是嘈杂的导数估计也会表现出基于随机梯度的优化的快速、稳定的收敛，正如我们通过热设计、扩散形状和计算机图形学的示例所展示的那样。