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).

对于展现时空动态响应的系统的演化进行预测是促进科学创新的关键技术，传统基于方程的方法需要大规模并行计算平台和大量的计算成本。相比之下，我们提出了一种名为 Latent Dynamics Network 的新型架构，该架构使用降维和深度学习算法来描述系统演化的低维潜在空间，从而预测空间依赖场对外部输入的时间演化，并在几个测试案例上验证了该方法的高效性和精确性。