Graph-Informed Neural Networks

Graph-Informed Neural Networks (GINNs) present a strategy for incorporating domain knowledge into scientific machine learning for complex physical systems. The construction utilizes probabilistic graphical models (PGMs) to incorporate expert knowledge, available data, constraints, etc. with physics-based models such as systems of ordinary and partial differential equations (ODEs and PDEs). Computationally intensive nodes in this hybrid model are replaced by the hidden nodes of a neural network (i.e., learned features). Once trained, the resulting GINN surrogate can cheaply generate physically-relevant predictions at scale thereby enabling robust sensitivity analysis and uncertainty quantification (UQ). As proof of concept, we build a GINN for a multiscale model of electrical double-layer capacitor dynamics embedded into a Bayesian network (BN) PDE hybrid model. In recent years, several approaches have been proposed to inform deep neural networks (DNNs) of physical laws and constraints to ensure they produce physically sound predictions. Two main classes of DNNs for building surrogate representations of physics-based models described by PDEs have emerged: physics-informed NNs (PINNs) (Raissi, Perdikaris, and Karniadakis 2019) and “data-free” physics-constrained NNs (Zhu et al. 2019). Our approach uses the well-known concept of PGMs to embed domain knowledge, including correlations between control variables (CVs), into standard DNNs by only modifying their input layer structure and enabling the use of a standard penalty in the loss function, e.g., `1 (lasso regression) or `2 (ridge regression) regularization. This non-intrusive approach permits the use of off-the-shelf software like TensorFlow or PyTorch with minimal effort from the user, while remaining compatible with PINNs and other customized NN architectures which can be used to replace individual computational bottlenecks in the physics-based representation. GINNs are particularly suited to enhance the computational workflow for complex systems featuring intrinsic computational bottlenecks and intricate physical relations among input CVs. Hence, to showcase the potential of this approach, we apply a GINN to simulation-based decisionmaking in electrical double-layer (EDL) supercapacitors, Copyright © 2021, for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CCBY 4.0). where it is deployed to build highly accurate kernel density estimators (KDEs) for the probability density functions (PDFs) of relevant output quantities of interest (QoIs).