求解对称偏微分方程的DeepSets及其衍生网络
DeepSets and their derivative networks for solving symmetric PDEs
摘要：
求解非线性偏微分方程（PDE）的机器学习方法是当前的研究热点，已有的各种算法在高维空间表现出了高效的数值逼近能力。在本文中，我们引入了一类对置换不变的偏微分方程，称之为对称偏微分方程。这些问题是普遍存在的，从宇宙学到量子力学，以及具有可交换支付的多资产市场中的期权定价/套期保值。我们的主要应用实际上来自平均场控制问题的粒子近似。我们基于某些类型的神经网络（称为PointNet和DeepSet（及其相关的衍生网络））设计深度学习算法，用于同时计算对称偏微分方程的解及其梯度的近似值。我们说明了PointNet/DeepSet网络与经典前馈网络相比的性能和准确性，并提供了我们算法的几个数值结果，用于平均场系统风险，均值方差问题和最小/最大线性二次McKean-Vlasov控制问题的例子。
英文摘要：
Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we introduce a class of PDEs that are invariant to permutations, and called symmetric PDEs. Such problems are widespread, ranging from cosmology to quantum mechanics, and option pricing/hedging in multi-asset market with exchangeable payoff. Our main application comes actually from the particles approximation of mean-field control problems. We design deep learning algorithms based on certain types of neural networks, named PointNet and DeepSet (and their associated derivative networks), for computing simultaneously an approximation of the solution and its gradient to symmetric PDEs. We illustrate the performance and accuracy of the PointNet/DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean-field systemic risk, mean-variance problem and a min/max linear quadratic McKean-Vlasov control problem.
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