ACEENs: Enhanced causality-enforced networks with probability-driven adaptive sampling for time-dependent PDEs

This paper presents a novel adaptive causality-enforced evolutional networks (ACEENs) framework for solving time-dependent partial differential equations (PDEs). Building upon the baseline CEENs, the framework introduces a probability-driven adaptive sampling strategy, which utilizes residual and gradient information from previous time steps to dynamically refine the mesh discretization in regions critical to solution accuracy. Two specialized implementations are developed: ACEENs I, driven solely by residuals, and ACEENs II, which additionally exploits gradient information for more comprehensive refinement. The ACEENs framework strictly preserves temporal causality. Numerical experiments on four standard problems demonstrate that both ACEENs variants significantly outperform the baseline model in accuracy. Furthermore, computational results based on the Burgers equation further demonstrate that the proposed adaptive sampling method can reduce the variance of the loss estimate, thereby improving training stability. This indicates that the method can effectively capture the steep gradients and transient characteristics of the solution, potentially opening up a new avenue for the numerical simulation of complex time-varying PDEs.