Turbo-Charging Deep Learning Methods for Partial Differential Equations

Solving partial differential equations (PDEs) is a frequent necessity in numerous domains, ranging from complex systems simulation to financial derivatives pricing and continuous-time optimisation tasks. The challenging nature of PDEs, especially in high dimensions or cases involving non-linearities, calls for robust, innovative solutions. This paper leverages a deep neural network methodology, utilizing differential operators and boundary conditions in tandem with sampling techniques and minimising distinct loss terms. The role of physics-inspired neural networks in this approach is also highlighted. Our primary proposition is a Bayesian interpretation, where we address the issue as a hierarchical multi-objective optimisation problem augmented with adaptive sampling. We also introduce a concept of ’curriculum learning,’ which parallels control variates, thereby facilitating further variance reduction and the re-utilisation of solutions derived from assorted problems. Our methods notably enhance the speed of convergence and diminish approximation errors. The effectiveness of our strategies is demonstrated through illustrative examples, solidifying their value in practical applications.