Towards fast weak adversarial training to solve high dimensional parabolic partial differential equations using XNODE-WAN

Due to the curse of dimensionality, solving high dimensional parabolic partial differential equations (PDEs) has been a challenging problem for decades. Recently, weak adversarial network (WAN) proposed in Zang et al. (2020) [17] offered flexible and computationally efficient approach tackle this defined on arbitrary domains by leveraging solution. WAN reformulates PDE as generative network, where solution (primal network) test function (adversarial are parameterized multi-layer deep neural networks (DNNs). However, it is not yet clear whether DNNs most effective model solutions they do take into account fundamentally different roles played time spatial variables To reinforce difference, we design novel so-called XNODE primal which built ODE (NODE) with additional dependency incorporate priori information PDEs serve universal approximation The hybrid method (XNODE-WAN), integrating within framework, leads significant improvement performance efficiency training. Numerical results show that our can reduce training fraction model.