Solving non-local nonlinear Partial Differential Equations in high dimensions with HighDimPDE.jl

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Abstract

Non-local nonlinear Partial Differential Equations arise in a variety of fields in Biology and are used for modelling e.g. morphogeneis, cancer evolution or gene regulatory networks. They are also present in many models of evolution, for example in population genetics with the non-local Fisher KPP equation, or in quantitative genetics where populations are structured with quantitative traits.

Non-local PDEs are difficult to solve and even more so given that the dimensionality of those problems can be high. In evolutionary dynamics, the dimension of the solution relates to the dimension of the strategy space. In biology, it coincides e.g. with the number of genes modelled, or with the dimension of the geographical space and/or the dimension of the phenotypic space that represent the population.

Here, we present two algorithms that have recently proved successful in solving such PDEs in high dimensions, namely full history recursive multilevel Picard approximations and deep learning based. We further introduce HighDimPDE.jl, a user friendly Julia library that impelements those algorithms, and present some showcase examples.

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