Non-linear partial differential equations, stochastic representations, and numerical approximation by deep learning

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Geometric differential equations, describing how certain surfaces evolve, emerge in stochastic control problems, for example motivated in mathematical finance. The worst-case model for a long-term investor exploiting market volatility turns out to be characterised by such an equation. Well-established methods are used to solve differential equations for which analytically explicit solutions are not known. One approach is to use numerical methods (for example, finite difference and finite element schemes). An alternative is to reformulate such equations in probabilistic terms and to use simulation methods (for example, Monte Carlo and dynamic programming). The first approach performs very well in small dimensions but suffers from the curse of dimensionality. The second approach tends to be easily implementable but is usually slow since each simulation provides the solution at only one point.In the recent few years new exciting research has been proposed (several researchers working on this field in and outside the UK are named below) to 'learn' the solution. To this end, the solution is approximated by a (shallow or deep) neural network, i.e., through a composition of several linear and non-linear functions. The justification of this method is an application of the so-called universal approximation results as a core idea. The neural network is 'trained' by standard backpropagation techniques. There are different proposals for the corresponding functional to be minimised; e.g. based on the minimisation of certain operator norms in the context of adversarial learning or using stochastic representations (e.g. via Feynman-Kac). The later approach, for example, has been very successfully applied to differential equations that can be associated to backward stochastic differential equations.For this project the PI plans to work on developing this approach to a large class of geometric differential equations. Due to their importance (e.g. in physics) these equations have been studied extensively in differential geometry and in the partial differential equations' literature. One important example is the mean curvature flow, which is described by these equations. The seminal work by Soner and Touzi connects these equations to stochastic control problems. Similarly, Kohn and Serfaty link them to the value functions of deterministic games between two players. In a similar spirit as Soner and Touzi, the PI and Larsson have recently established a relationship between a certain class of these differential equations (especially those ones de- scribing the so-called 'minimum curvature' flow) to a specific control problem that describes the time a martingale can be contained in a compact set.These links between geometric differential equations and stochastic control problems (and also deterministic games) open up a promising avenue to find good numerical approximations to such high-dimensional problems. The PI plans to exploit these links to design and study the use of neural networks as a numerical approximation to geometric differential equations, starting with the minimum curvature flow.The project has an experimental component that implements the algorithms and provides insights in the speed of convergence, the necessary number of layers, learning rates, other hyperparameters, and the effect of exploiting symmetries (in the domain of the differential equation). This component also yields a proof-of-concept and will be made publicly availably via a GitHub repository. The second component of the project establishes expression rates and additional properties rigorously.