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:预测对外部刺激呈现时空动态的系统的演变是促进科学创新的关键技术。传统的基于方程的方法利用第一原理通过高维微分方程组的数值近似来产生预测,因此需要大规模的并行计算平台,并需要大量的计算成本。相反,数据驱动的方法通过利用降维和深度学习算法来在低维潜在空间中描述系统演变。我们提出了一种新颖的体系结构,名为潜在动力学网络(LDNet),能够发现潜在的低维动力学,以此预测受外部输入影响的空间依赖场的时间演变。与流行的方法不同,在这些方法中,通过自编码器将系统状态的高维离散化映射到其本身的潜在表示上学习解流形,而LDNets在学习潜在动力学时自动发现低维流形,而不会在高维空间中操作。此外,LDNets是无网格算法,不需要在预先确定的点网格上重构输出,而是在域中任何点处进行重构,从而实现跨查询点的权重共享。这些特性使LDNets轻量化且易于训练,具有出色的精度和泛化性能,即使在时间外推情况下也是如此。我们在几个测试案例上验证了我们的方法,并且显示,在一个具有挑战性的高度非线性问题上,LDNets表现优于现有技术,其准确性(标准化误差约为5倍较小)通过使用数量更少的可训练参数(超过10倍)实现。