While the popularity of physics-informed neural networks (PINNs) is steadily
rising, to this date PINNs have not been successful in simulating dynamical
systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In
this work we attribute this shortcoming to the inability of existing PINNs
formulations to respect the spatio-temporal causal structure that is inherent
to the evolution of physical systems. We argue that this is a fundamental
limitation and a key source of error that can ultimately steer PINN models to
converge towards erroneous solutions. We address this pathology by proposing a
simple re-formulation of PINNs loss functions that can explicitly account for
physical causality during model training. We demonstrate that this simple
modification alone is enough to introduce significant accuracy improvements, as
well as a practical quantitative mechanism for assessing the convergence of a
PINNs model. We provide state-of-the-art numerical results across a series of
benchmarks for which existing PINNs formulations fail, including the chaotic
Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the
Navier-Stokes equations in the turbulent regime. To the best of our knowledge,
this is the first time that PINNs have been successful in simulating such
systems, introducing new opportunities for their applicability to problems of
industrial complexity.

本文提出一种简单的 PINNs 卷积方法，可显式考虑物理因果关系，在多尺度、混沌或湍流行为的动力学系统中实现更高的准确性和可靠性。