Predicting the evolution of systems that exhibit spatio-temporal dynamics in response to external stimuli is a key enabling technology fostering scientific innovation. Traditional equations-based approaches leverage first principles to yield predictions through the numerical approximation of high-dimensional systems of differential equations, thus calling for large-scale parallel computing platforms and requiring large computational costs. Data-driven approaches, instead, enable the description of systems evolution in low-dimensional latent spaces, by leveraging dimensionality reduction and deep learning algorithms. We propose a novel architecture, named Latent Dynamics Network (LDNet), which is able to discover low-dimensional intrinsic dynamics of possibly non-Markovian dynamical systems, thus predicting the time evolution of space-dependent fields in response to external inputs. Unlike popular approaches, in which the latent representation of the solution manifold is learned by means of auto-encoders that map a high-dimensional discretization of the system state into itself, LDNets automatically discover a low-dimensional manifold while learning the latent dynamics, without ever operating in the high-dimensional space. Furthermore, LDNets are meshless algorithms that do not reconstruct the output on a predetermined grid of points, but rather at any point of the domain, thus enabling weight-sharing across query-points. These features make LDNets lightweight and easy-to-train, with excellent accuracy and generalization properties, even in time-extrapolation regimes. We validate our method on several test cases and we show that, for a challenging highly-nonlinear problem, LDNets outperform state-of-the-art methods in terms of accuracy (normalized error 5 times smaller), by employing a dramatically smaller number of trainable parameters (more than 10 times fewer).

翻译：预测系统在外部刺激下呈现时空动态演化的能力，是推动科学创新的关键使能技术。传统基于方程的方法利用第一性原理，通过对高维微分方程系统的数值近似进行预测，这需要大规模并行计算平台并耗费大量计算成本。相比之下，数据驱动方法通过利用降维和深度学习算法，能够在低维潜空间中描述系统的演化。我们提出了一种名为潜动力网络（LDNet）的新型架构，能够发现可能非马尔可夫动力系统的低维内在动力学，从而预测空间依赖场在外部输入作用下的时间演化。与主流方法通过自编码器将系统状态的高维离散化映射到自身来学习解流形的潜表示不同，LDNets在无需操作高维空间的情况下，自动发现低维流形的同时学习潜在动力学。此外，LDNets是一种无网格算法，无需在预定网格点上重建输出，而是可以在定义域内任意点进行重建，从而实现对查询点的权重共享。这些特性使LDNets轻量化且易于训练，即使在外推时间域中也具有出色的精度和泛化性能。我们在多个测试案例中验证了该方法，并证明：对于一个极具挑战性的高度非线性问题，LDNets通过使用显著更少的可训练参数（减少超过10倍），在精度上（归一化误差缩小5倍）超越了现有最优方法。