Rui Wang

Rui Wang, Yihe Dong, Sercan Ö. Arik et al.

Temporal distributional shifts, with underlying dynamics changing over time, frequently occur in real-world time series and pose a fundamental challenge for deep neural networks (DNNs). In this paper, we propose a novel deep sequence model based on the Koopman theory for time series forecasting: Koopman Neural Forecaster (KNF) which leverages DNNs to learn the linear Koopman space and the coefficients of chosen measurement functions. KNF imposes appropriate inductive biases for improved robustness against distributional shifts, employing both a global operator to learn shared characteristics and a local operator to capture changing dynamics, as well as a specially-designed feedback loop to continuously update the learned operators over time for rapidly varying behaviors. We demonstrate that \ours{} achieves superior performance compared to the alternatives, on multiple time series datasets that are shown to suffer from distribution shifts.

Rui Wang, Robin Walters, Rose Yu

Recent work has shown deep learning can accelerate the prediction of physical dynamics relative to numerical solvers. However, limited physical accuracy and an inability to generalize under distributional shift limit its applicability to the real world. We propose to improve accuracy and generalization by incorporating symmetries into convolutional neural networks. Specifically, we employ a variety of methods each tailored to enforce a different symmetry. Our models are both theoretically and experimentally robust to distributional shift by symmetry group transformations and enjoy favorable sample complexity. We demonstrate the advantage of our approach on a variety of physical dynamics including Rayleigh Bénard convection and real-world ocean currents and temperatures. Compared with image or text applications, our work is a significant step towards applying equivariant neural networks to high-dimensional systems with complex dynamics. We open-source our simulation, data, and code at \url{https://github.com/Rose-STL-Lab/Equivariant-Net}.

Jing Wang, Zheng Li, Pengyu Lai et al.

Multiscale phenomena manifest across various scientific domains, presenting a ubiquitous challenge in accurately and effectively simulating multiscale dynamics in complex systems. In this paper, a novel decoupling solving paradigm is proposed through modelling large-scale dynamics independently and treating small-scale dynamics as a slaved system. A Spectral Physics-informed Neural Network (PINN) is developed to characterize the small-scale system in an efficient and accurate way, addressing the challenges posed by the representation of multiscale dynamics in neural networks. The effectiveness of the method is demonstrated through extensive numerical experiments, including one-dimensional Kuramot-Sivashinsky equation, two- and three-dimensional Navier-Stokes equations, showcasing its versatility in addressing problems of fluid dynamics. Furthermore, we also delve into the application of the proposed approach to more complex problems, including non-uniform meshes, complex geometries, large-scale data with noise, and high-dimensional small-scale dynamics. The discussions about these scenarios contribute to a comprehensive understanding of the method's capabilities and limitations. By enabling the acquisition of large-scale data with minimal computational demands, coupled with the efficient and accurate characterization of small-scale dynamics via Spectral PINN, our approach offers a valuable and promising approach for researchers seeking to tackle multiscale phenomena effectively.

Rui Wang, Robin Walters, Rose Yu

Incorporating symmetry as an inductive bias into neural network architecture has led to improvements in generalization, data efficiency, and physical consistency in dynamics modeling. Methods such as CNNs or equivariant neural networks use weight tying to enforce symmetries such as shift invariance or rotational equivariance. However, despite the fact that physical laws obey many symmetries, real-world dynamical data rarely conforms to strict mathematical symmetry either due to noisy or incomplete data or to symmetry breaking features in the underlying dynamical system. We explore approximately equivariant networks which are biased towards preserving symmetry but are not strictly constrained to do so. By relaxing equivariance constraints, we find that our models can outperform both baselines with no symmetry bias and baselines with overly strict symmetry in both simulated turbulence domains and real-world multi-stream jet flow.

Rui Wang, Rose Yu

Modeling complex physical dynamics is a fundamental task in science and engineering. Traditional physics-based models are sample efficient, and interpretable but often rely on rigid assumptions. Furthermore, direct numerical approximation is usually computationally intensive, requiring significant computational resources and expertise, and many real-world systems do not have fully-known governing laws. While deep learning (DL) provides novel alternatives for efficiently recognizing complex patterns and emulating nonlinear dynamics, its predictions do not necessarily obey the governing laws of physical systems, nor do they generalize well across different systems. Thus, the study of physics-guided DL emerged and has gained great progress. Physics-guided DL aims to take the best from both physics-based modeling and state-of-the-art DL models to better solve scientific problems. In this paper, we provide a structured overview of existing methodologies of integrating prior physical knowledge or physics-based modeling into DL, with a special emphasis on learning dynamical systems. We also discuss the fundamental challenges and emerging opportunities in the area.

Rui Wang, Danielle Maddix, Christos Faloutsos et al.

How can we learn a dynamical system to make forecasts, when some variables are unobserved? For instance, in COVID-19, we want to forecast the number of infected and death cases but we do not know the count of susceptible and exposed people. While mechanics compartment models are widely used in epidemic modeling, data-driven models are emerging for disease forecasting. We first formalize the learning of physics-based models as AutoODE, which leverages automatic differentiation to estimate the model parameters. Through a benchmark study on COVID-19 forecasting, we notice that physics-based mechanistic models significantly outperform deep learning. Our method obtains a 57.4% reduction in mean absolute errors for 7-day ahead COVID-19 forecasting compared with the best deep learning competitor. Such performance differences highlight the generalization problem in dynamical system learning due to distribution shift. We identify two scenarios where distribution shift can occur: changes in data domain and changes in parameter domain (system dynamics). Through systematic experiments on several dynamical systems, we found that deep learning models fail to forecast well under both scenarios. While much research on distribution shift has focused on changes in the data domain, our work calls attention to rethink generalization for learning dynamical systems.

Rui Wang, Karthik Kashinath, Mustafa Mustafa et al.

While deep learning has shown tremendous success in a wide range of domains, it remains a grand challenge to incorporate physical principles in a systematic manner to the design, training, and inference of such models. In this paper, we aim to predict turbulent flow by learning its highly nonlinear dynamics from spatiotemporal velocity fields of large-scale fluid flow simulations of relevance to turbulence modeling and climate modeling. We adopt a hybrid approach by marrying two well-established turbulent flow simulation techniques with deep learning. Specifically, we introduce trainable spectral filters in a coupled model of Reynolds-averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES), followed by a specialized U-net for prediction. Our approach, which we call turbulent-Flow Net (TF-Net), is grounded in a principled physics model, yet offers the flexibility of learned representations. We compare our model, TF-Net, with state-of-the-art baselines and observe significant reductions in error for predictions 60 frames ahead. Most importantly, our method predicts physical fields that obey desirable physical characteristics, such as conservation of mass, whilst faithfully emulating the turbulent kinetic energy field and spectrum, which are critical for accurate prediction of turbulent flows.