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Physics-Incorporated Convolutional Recurrent Neural Networks For Physics-Incorporated Convolutional Recurrent Neural Networks for Source Identification and Forecasting of Dynamical Systems < Priyabrata Saha , Saurabh Dash and Saibal Mukhopadhyay School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA ARTICLEINFO ABSTRACT Keywords: Spatio-temporal dynamics of physical processes are generally modeled using partial differential equa- Dynamical systems tions (PDEs). Though the core dynamics follows some principles of physics, real-world physical pro- Partial differential equation cesses are often driven by unknown external sources. In such cases, developing a purely analytical Recurrent neural networks model becomes very difficult and data-driven modeling can be of assistance. In this paper, we present Physics-incorporated neural networks a hybrid framework combining physics-based numerical models with deep learning for source identifi- cation and forecasting of spatio-temporal dynamical systems with unobservable time-varying external sources. We formulate our model PhICNet as a convolutional recurrent neural network (RNN) which is end-to-end trainable for spatio-temporal evolution prediction of dynamical systems and learns the source behavior as an internal state of the RNN. Experimental results show that the proposed model can forecast the dynamics for a relatively long time and identify the sources as well. 1. Introduction a vertical displacement on the membrane where it is applied and the membrane will respond to restore its shape due to Understanding the behavior of dynamical systems is a its elasticity. Consequently, new distortions/undulations will fundamental problem in science and engineering. Classi- be created in the neighboring region forming a wave that cal approaches of modeling dynamical systems involve for- propagates across the membrane. Consider the case where mulating ordinary or partial differential equations (ODEs or the undulation of the membrane can be observed as a regu- PDEs) based on various physical principles, profound rea- larly sampled spatiotemporal sequence but the spatiotempo- soning, intuition, knowledge and verifying those with exper- ral variation in the source term is not observable. Moreover, iments and observations. However, in many modern fields, though the basic governing dynamics (wave equation) of the such as climate science, neuroscience, epidemiology, finance system is known, the physical parameters such as propaga- etc., analytical modeling is very challenging and/or can only tion speed in that particular medium may not be known. In describe the system behavior partially. Recent successes of such scenarios, we need to be able to predict spatiotempo- machine learning methods in complex sequence prediction ral evolution of dynamical systems from partial knowledge tasks [30,4, 34,5,9, 10, 18] along with development in sen- of the governing dynamics and limited observability of the sor technologies and computing systems motivate to predict factors that influence the system behaviors. the evolution of dynamical system directly from observation In this paper, we consider to model a generic PDE-based data. Consequently, a number of machine learning mod- dynamical system which is perturbed with external sources els which incorporate knowledge from physics or applied that follow another independent dynamics. Our goal is to de- mathematics, have been introduced for modeling complex sign a neural network model that can be used for long-term dynamical systems [27, 25, 21, 15, 13,1]. prediction of the entire systems, as well as of the source dy- Real-world dynamical systems are often subjected to per- namics. We assume the physical quantity that follows the turbation from time-varying external sources. For example, combined dynamics can be observed as regularly sampled in many fields of applied sciences, such as ocean acoustics, arXiv:2004.06243v3 [cs.LG] 31 Aug 2021 spatiotemporal sequence, but the source or perturbation is geophysics etc., the dynamical systems are often stimulated not observable (neither during training nor testing) sepa- with external sources to study the corresponding environ- rately. It is further assumed that we know what type of sys- ment. Perturbation from an external source, generally re- tem we are observing and therefore scientific knowledge of ferred as source term, acts as an external force or stimula- such system can be incorporated in the model. In particu- tion that changes the behavior of the system from its default lar, we assume that the analytical form the underlying PDE course. Figure1(a) shows an example where an elastic mem- is known a priori, but the physical parameters (e.g., propa- brane under tension is being perturbed with an independent gation speed in the elastic membrane example) of the system time-varying pressure. The perturbation pressure will cause are unknown. Therefore, it is not possible to separate out the ? This material is based on work sponsored by the Army Research Of- source dynamics from the PDE. Hence, the main challenge fice and was accomplished under Grant Number W911NF-19-1-0447. The lies in the fact that the unknown source dynamics cannot be views and conclusions contained in this document are those of the authors learned using direct supervision because we do not have ac- and should not be interpreted as representing the official policies, either ex- pressed or implied, of the Army Research Office or the U.S. Government. cess to the ground truth. Rather, we need to formulate a < Corresponding author supervised learning task for the (observable) combined dy- [email protected] (P. Saha) namics that will inherently learn the hidden source dynam- 0000-0002-6933-0660 ORCID(s): (P. Saha) ics. P Saha et al.: Preprint submitted to Elsevier Page 1 of 15 RNNs for Source Identication and Forecasting of Dynamical Systems 푈෡푡+1 Unknown source dynamics time Data-driven Perturbation (unobservable) states Internal Physics-based PhICNet Cell Observed map (푈푡) 푈푡 (a) (b) Figure 1: (a): An example dynamical system governed by partially-known PDE and unknown source dynamics. (b): High-level diagram of the recurrent network cell that is used to model such dynamics. Unobservable perturbation is learned as an internal state of PhICNet cell. We propose Physics-Incorporated Convolutional Recur- PDE-Net [15] to incorporate nonlinear source term; however rent Neural Networks (PhICNet) that combine physical mod- they consider only internal source term that is a nonlinear els with data-driven models to learn the behavior of dynami- function of the observed physical quantity. DHPM, PDE- cal systems with unobservable time-varying external source Net only consider PDEs that are first-order in time, although term. Figure1(b) shows the high-level diagram of a PhICNet can be extended to model higher temporal order systems if cell. The basic concept of PhICnet is based on two key con- we know the temporal order a priori. tributions. First, a generic homogeneous PDE of any tem- Our approach fundamentally differs from the past ap- poral order can be mapped into a recurrent neural network proaches as we model the system as an RNN that couples (RNN) structure with trainable physical parameters. Sec- the known and unknown dynamics within the hidden cells ond, the basic RNN structure for homogeneous PDE can be and enables end-to-end training. We evaluate our model modified and integrated with a residual encoder-decoder net- along with other baselines for three types of dynamical sys- work to identify and learn the source dynamics. The RNN tems: a heat diffusion system, a wave propagation system structure stores the homogeneous solution of the underlying and, a Burgers’ fluid flow system. Experiments show that PDE as an internal state which is then compared with the in- our model provides more accurate prediction compared to put (observation) at the next step to find out if there exists the baseline models. Furthermore, we show that our model some source term or perturbation. The residual encoder- can predict the source dynamics separately which is not pos- decoder network learns to propagate the source term as it sible with other models. is in case of constant perturbation or predict its progress in case of dynamic perturbation. The PhICNet cell stores 2. Related Work the estimated perturbation or source term as an internal cell 2.1. RNNs for Dynamical Systems which can be used to understand behaviour of source dy- namics. The integrated model can be trained end-to-end us- Several studies have interpreted RNNs as approximation ing stochastic gradient descent (SGD) based backpropaga- of dynamical systems [6, 11, 12,3]. Recently, a number tion through time (BPTT) algorithm. of RNN architectures have been proposed for data-driven Few existing models in literature can be used or extended modeling of dynamical systems. Trischler and D’Eleuterio to perform the spatiotemporal sequence prediction of dy- [31] proposed an algorithm for efficiently training RNNs to namical systems with time-varying independent source. Pure replicate dynamical systems and demonstrated its capabil- data-driven models like ConvLSTM [34], residual networks ity to approximate attractor dynamical systems. A class of [7, 17] can be used directly, but these models lack consid- RNNs, namely Tensor-RNNs, has been proposed in [36, 37] eration of underlying physical dynamics resulting in limited for long-term prediction of nonlinear dynamical systems. Yeo accuracy. Furthermore, these pure data-driven models can- and Melnyk [35] use LSTM
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