Multifidelity domain decomposition-based physics-informed neural networks for time-dependent problems

Multiscale problems are challenging for neural network-based discretizations of differential equations, such as physics-informed neural networks (PINNs). This can be (partly) attributed to the so-called spectral bias of neural networks. To improve the performance of PINNs for time-dependent problems, a combination of multifidelity stacking PINNs and domain decomposition-based finite basis PINNs are employed. In particular, to learn the high-fidelity part of the multifidelity model, a domain decomposition in time is employed. The performance is investigated for a pendulum and a two-frequency problem as well as the Allen-Cahn equation. It can be observed that the domain decomposition approach clearly improves the PINN and stacking PINN approaches.

A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modeling

One of the most challenging and consequential problems in climate modeling is to provide probabilistic projections of sea level rise. A large part of the uncertainty of sea level projections is due to uncertainty in ice sheet dynamics. At the moment, accurate quantification of the uncertainty is hindered by the cost of ice sheet computational models. In this work, we develop a hybrid approach to approximate existing ice sheet computational models at a fraction of their cost. Our approach consists of replacing the finite element model for the momentum equations for the ice velocity, the most expensive part of an ice sheet model, with a Deep Operator Network, while retaining a classic finite element discretization for the evolution of the ice thickness. We show that the resulting hybrid model is very accurate and it is an order of magnitude faster than the traditional finite element model. Further, a distinctive feature of the proposed model compared to other neural network approaches, is that it can handle high-dimensional parameter spaces (parameter fields) such as the basal friction at the bed of the glacier, and can therefore be used for generating samples for uncertainty quantification. We study the impact of hyper-parameters, number of unknowns and correlation length of the parameter distribution on the training and accuracy of the Deep Operator Network on a synthetic ice sheet model. We then target the evolution of the Humboldt glacier in Greenland and show that our hybrid model can provide accurate statistics of the glacier mass loss and can be effectively used to accelerate the quantification of uncertainty.

Multifidelity Deep Operator Networks

Operator learning for complex nonlinear operators is increasingly common in modeling physical systems. However, training machine learning methods to learn such operators requires a large amount of expensive, high-fidelity data. In this work, we present a composite Deep Operator Network (DeepONet) for learning using two datasets with different levels of fidelity, to accurately learn complex operators when sufficient high-fidelity data is not available. Additionally, we demonstrate that the presence of low-fidelity data can improve the predictions of physics-informed learning with DeepONets.

Physics-informed CoKriging model of a redox flow battery

Redox flow batteries (RFBs) offer the capability to store large amounts of energy cheaply and efficiently, however, there is a need for fast and accurate models of the charge-discharge curve of a RFB to potentially improve the battery capacity and performance. We develop a multifidelity model for predicting the charge-discharge curve of a RFB. In the multifidelity model, we use the Physics-informed CoKriging (CoPhIK) machine learning method that is trained on experimental data and constrained by the so-called "zero-dimensional" physics-based model. Here we demonstrate that the model shows good agreement with experimental results and significant improvements over existing zero-dimensional models. We show that the proposed model is robust as it is not sensitive to the input parameters in the zero-dimensional model. We also show that only a small amount of high-fidelity experimental datasets are needed for accurate predictions for the range of considered input parameters, which include current density, flow rate, and initial concentrations.

Learning Unknown Physics of non-Newtonian Fluids

We extend the physics-informed neural network (PINN) method to learn viscosity models of two non-Newtonian systems (polymer melts and suspensions of particles) using only velocity measurements. The PINN-inferred viscosity models agree with the empirical models for shear rates with large absolute values but deviate for shear rates near zero where the analytical models have an unphysical singularity. Once a viscosity model is learned, we use the PINN method to solve the momentum conservation equation for non-Newtonian fluid flow using only the boundary conditions.