Sergey Oladyshkin

Coşku Can Horuz, Matthias Karlbauer, Timothy Praditia et al.

Spatiotemporal partial differential equations (PDEs) find extensive application across various scientific and engineering fields. While numerous models have emerged from both physics and machine learning (ML) communities, there is a growing trend towards integrating these approaches to develop hybrid architectures known as physics-aware machine learning models. Among these, the finite volume neural network (FINN) has emerged as a recent addition. FINN has proven to be particularly efficient in uncovering latent structures in data. In this study, we explore the capabilities of FINN in tackling the shallow-water equations, which simulates wave dynamics in coastal regions. Specifically, we investigate FINN's efficacy to reconstruct underwater topography based on these particular wave equations. Our findings reveal that FINN exhibits a remarkable capacity to infer topography solely from wave dynamics, distinguishing itself from both conventional ML and physics-aware ML models. Our results underscore the potential of FINN in advancing our understanding of spatiotemporal phenomena and enhancing parametrization capabilities in related domains.

Nils Wildt, Daniel M. Tartakovsky, Sergey Oladyshkin et al.

Ordinary differential equations (ODEs) are a conventional way to describe the observed dynamics of physical systems. Scientists typically hypothesize about dynamical behavior, propose a mathematical model, and compare its predictions to data. However, modern computing and algorithmic advances now enable purely data-driven learning of governing dynamics directly from observations. In data-driven settings, one learns the ODE's right-hand side (RHS). Dense measurements are often assumed, yet high temporal resolution is typically both cumbersome and expensive. Consequently, one usually has only sparsely sampled data. In this work we introduce ChaosODE (CODE), a Polynomial Chaos ODE Expansion in which we use an arbitrary Polynomial Chaos Expansion (aPCE) for the ODE's right-hand side, resulting in a global orthonormal polynomial representation of dynamics. We evaluate the performance of CODE in several experiments on the Lotka-Volterra system, across varying noise levels, initial conditions, and predictions far into the future, even on previously unseen initial conditions. CODE exhibits remarkable extrapolation capabilities even when evaluated under novel initial conditions and shows advantages compared to well-examined methods using neural networks (NeuralODE) or kernel approximators (KernelODE) as the RHS representer. We observe that the high flexibility of NeuralODE and KernelODE degrades extrapolation capabilities under scarce data and measurement noise. Finally, we provide practical guidelines for robust optimization of dynamics-learning problems and illustrate them in the accompanying code.

Matthias Karlbauer, Timothy Praditia, Sebastian Otte et al.

We introduce a compositional physics-aware FInite volume Neural Network (FINN) for learning spatiotemporal advection-diffusion processes. FINN implements a new way of combining the learning abilities of artificial neural networks with physical and structural knowledge from numerical simulation by modeling the constituents of partial differential equations (PDEs) in a compositional manner. Results on both one- and two-dimensional PDEs (Burgers', diffusion-sorption, diffusion-reaction, Allen--Cahn) demonstrate FINN's superior modeling accuracy and excellent out-of-distribution generalization ability beyond initial and boundary conditions. With only one tenth of the number of parameters on average, FINN outperforms pure machine learning and other state-of-the-art physics-aware models in all cases -- often even by multiple orders of magnitude. Moreover, FINN outperforms a calibrated physical model when approximating sparse real-world data in a diffusion-sorption scenario, confirming its generalization abilities and showing explanatory potential by revealing the unknown retardation factor of the observed process.

Timothy Praditia, Matthias Karlbauer, Sebastian Otte et al.

Data-driven modeling of spatiotemporal physical processes with general deep learning methods is a highly challenging task. It is further exacerbated by the limited availability of data, leading to poor generalizations in standard neural network models. To tackle this issue, we introduce a new approach called the Finite Volume Neural Network (FINN). The FINN method adopts the numerical structure of the well-known Finite Volume Method for handling partial differential equations, so that each quantity of interest follows its own adaptable conservation law, while it concurrently accommodates learnable parameters. As a result, FINN enables better handling of fluxes between control volumes and therefore proper treatment of different types of numerical boundary conditions. We demonstrate the effectiveness of our approach with a subsurface contaminant transport problem, which is governed by a non-linear diffusion-sorption process. FINN does not only generalize better to differing boundary conditions compared to other methods, it is also capable to explicitly extract and learn the constitutive relationships (expressed by the retardation factor). More importantly, FINN shows excellent generalization ability when applied to both synthetic datasets and real, sparse experimental data, thus underlining its relevance as a data-driven modeling tool.