An overview on deep learning-based approximation methods for partial differential equations

Partial differential equations (PDEs) are ubiquitous in mathematics as tools for modelling processes in nature or in man-made complex systems. PDEs appear, e.g., as Hamilton-Jacobi-Bellman equations in optimal control problems for describing the value function associated to the control problem, as Zakai or Kushner equations in nonlinear filtering problems for describing the conditional probability distribution of the state of the signal process in the nonlinear filtering problem, in models for the approximative valuation of financial products such as financial derivative contracts, and in the approximative description of the distribution of species in ecosystems to model biodiversity under changing climate conditions. The PDEs which appear in the abovenamed applications are often nonlinear and high-dimensional where, e.g., in the case of optimal control problems, the PDE dimension d N = {1,2,3,...} corresponds to the number of agents, particles, or resources in the optimal control problem, where, e.g., in the case of the approximative valuation of financial products, the PDE dimension d N corresponds to the number of financial assets (such as stocks, commodities, exchange rates, and interest rates) in the involved hedging portfolio, and where, e.g., in the case of the approximative description of the distribution of species in ecosystems, the PDE dimension d N corresponds to the number of characteristic traits of the species in the ecosystem under consideration. High-dimensional nonlinear PDEs cannot be solved analytically in nearly all cases and it is one of the most challenging issues in applied mathematics to design and analyze approximation methods for high-dimensional nonlinear PDEs.