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Error estimates for physics informed neural networks approximating the Navier-Stokes equations
(2022)SAM Research ReportWe prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with (extended) physics informed neural networks. We show that the underlying PDE residual can be made arbitrarily small for tanh neural networks with two hidden layers. Moreover, the total error can be estimated in terms of the training error, network size and number of quadrature points. The theory is illustrated with ...Report -
On the discrete equation model for compressible multiphase fluid flows
(2022)SAM Research ReportThe modeling of multi-phase flow is very challenging, given the range of scales as well as the diversity of flow regimes that one encounters in this context. We revisit the discrete equation method (DEM) for two-phase flow in the absence of heat conduction and mass transfer. We analyze the resulting probability coefficients and prove their local convexity, rigorously establishing that our version of DEM can model different flow regimes ...Report -
Agnostic Physics-Driven Deep Learning
(2022)SAM Research ReportThis work establishes that a physical system can perform statistical learning without gradient computations, via an \emph{Agnostic Equilibrium Propagation} (AEqprop) procedure that combines energy minimization, homeostatic control, and nudging towards the correct response. In AEqprop, the specifics of the system do not have to be known: the procedure is based only on external manipulations, and produces a stochastic gradient descent without ...Report -
Variable-Input Deep Operator Networks
(2022)SAM Research ReportExisting architectures for operator learning require that the number and locations of sensors (where the input functions are evaluated) remain the same across all training and test samples, significantly restricting the range of their applicability. We address this issue by proposing a novel operator learning framework, termed Variable-Input Deep Operator Network (VIDON), which allows for random sensors whose number and locations can vary ...Report -
Generic bounds on the approximation error for physics-informed (and) operator learning
(2022)SAM Research ReportWe propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution operator of generic partial differential equations ...Report -
Coupled Oscillatory Recurrent Neural Network (coRNN): An accurate and (gradient) stable architecture for learning long time dependencies
(2020)SAM Research ReportCircuits of biological neurons, such as in the functional parts of the brain can be modeled as networks of coupled oscillators. Inspired by the ability of these systems to express a rich set of outputs while keeping (gradients of) state variables bounded, we propose a novel architecture for recurrent neural networks. Our proposed RNN is based on a time-discretization of a system of second-order ordinary differential equations, modeling ...Report -
Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequences
(2020)SAM Research ReportWe propose a deep supervised learning algorithm based on low-discrepancy sequences as the training set. By a combination of theoretical arguments and extensive numerical experiments we demonstrate that the proposed algorithm significantly outperforms standard deep learning algorithms that are based on randomly chosen training data, for problems in moderately high dimensions. The proposed algorithm provides an efficient method for building ...Report -
Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs
(2020)SAM Research ReportPhysics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of ...Report -
Physics Informed Neural Networks for Simulating Radiative Transfer
(2020)SAM Research ReportWe propose a novel machine learning algorithm for simulating radiative transfer. Our algorithmis based on physics informed neural networks (PINNs), which are trained by minimizing the residualof the underlying radiative tranfer equations. We present extensive experiments and theoretical errorestimates to demonstrate that PINNs provide a very easy to implement, fast, robust and accuratemethod for simulating radiative transfer. We also ...Report -
Gradient Gating for Deep Multi-Rate Learning on Graphs
(2022)SAM Research ReportWe present Gradient Gating (G2), a novel framework for improving the performance of Graph Neural Networks (GNNs). Our framework is based on gating the output of GNN layers with a mechanism for multi-rate flow of message passing information across nodes of the underlying graph. Local gradients are harnessed to further modulate message passing updates. Our framework flexibly allows one to use any basic GNN layer as a wrapper around which ...Report