ESCP Europe

Langue

English

Publication year

15/03/2026

The Monge-Ampère equation is originated from geometric surface theory and is widely applied in optimal transport theory, image processing, optimization problem and so on. The numerical solution of the Monge-Ampère equation has recently attracted more and more attention. Physics-informed neural networks (PINNs), a new paradigm in numerical methods, introduce physical constraints during the training process so that the model not only can learn patterns in the data, but also satisfy the laws of physics. In our work, we try to solve the Monge-Ampère equation with Dirichlet boundary conditions by using the PINNs. To our knowledge, this is the first time that PINNs is applied to solve the Monge-Ampère equation. Unfortunately, the Monge-Ampère equation involves determinant calculation, which leads to calculation failure using the conventional PINNs. For this reason, inspired by the fixed-point method, we construct a Poisson series physics-informed neural networks (PS-PINNs) framework to solve this problem. The Monge-Ampère equation is transformed into a Poisson series using the fixed-point method, which avoids the direct computation of the determinant. As part of our analysis, we prove the convergence of loss function and neural networks in PS-PINNs. Moreover, we study the performance of PS-PINNs with source functions containing singularities and noise, as well as in asymmetric domains. It is worth noting that we can obtain better numerical results using a small number of sampling points and iterations. The data and code accompanying this paper are publicly available at https://github.com/RuiboZhangping/PSPINN.