Machine Learning for semi linear PDEs

Based on the resolution of the BSDE associated to the PDE first exhibited in [PP90] and using the time discretization scheme proposed in [BT04], some effective algorithms based on regressions manage to solve non linear PDEs in dimension above 4 (see [G 05; L 06]). As shown in [GT16] this technique is the source of a lot of research. Among others, we may refer to [FTW11] which generalizes this technique to full non linear equations by using the Second Order Backward Equation framework proposed in [Che 07]. This regression technique uses some basis functions that can be either some global polynomials as in [LS01] or some local polynomials as proposed in [BW12]: therefore this methodology still faces the curse of dimensionality and can only solve some problems in dimension below 7 or 8. Recently, [Hen 16; Bou 17; BTW17; War17] proposed to solve high dimensional PDE by using a branching method and a time step randomization applied to the Feyman-Kac representation of the PDE. In the case of semi-linear PDE's, a differentiation technique using some Malliavin weights as proposed in [Fou 99] allows to estimate the gradient Du of the solution. Unfortunately, branching techniques are only limited to small maturities, some small non-linearities and mainly to non-linearities that are polynomial in u and Du.