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This paper proposes the Nerual Energy Descent (NED) via neural network evolution equations for a wide class of deep learning problems. We show that deep learning can be reformulated as the evolution of network parameters in an evolution…
In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of infinitely many normal mode solutions to the…
We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
In the report, we propose a family of variable time-stepping ensemble algorithms for solving multiple incompressible Navier-Stokes equations (NSE) at one pass. The one-leg, two-step methods designed by Dahlquist, Liniger, and Nevanlinna…
Deep neural networks (NNs) encounter scalability limitations when confronted with a vast array of neurons, thereby constraining their achievable network depth. To address this challenge, we propose an integration of tensor networks (TN)…
Training deep neural networks is a highly nontrivial task, involving carefully selecting appropriate training algorithms, scheduling step sizes and tuning other hyperparameters. Trying different combinations can be quite labor-intensive and…
Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that…
Our proposed deeply-supervised nets (DSN) method simultaneously minimizes classification error while making the learning process of hidden layers direct and transparent. We make an attempt to boost the classification performance by studying…
Nonlinear feedback design via state-dependent Riccati equations is well established but unfeasible for large-scale systems because of computational costs. If the system can be embedded in the class of linear parameter-varying (LPV) systems…
Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional…
Dahlquist, Liniger, and Nevanlinna design a family of one-leg, two-step methods (the DLN method) that is second order, A- and G-stable for arbitrary, non-uniform time steps. Recently, the implementation of the DLN method can be simplified…
In many mechanistic medical, biological, physical and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs) can make simulations impractically slow. Biological models require the…
We propose a new per-layer adaptive step-size procedure for stochastic first-order optimization methods for minimizing empirical loss functions in deep learning, eliminating the need for the user to tune the learning rate (LR). The proposed…
In this work we present an application of modern deep learning methodologies to the numerical solution of partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into…
Navier-Stokes equations are well known in modelling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the…
In this paper we present a method to derive classical solutions of the Navier-Stokes equations for non-stationary initial value problems in domain $\mathbb{R}^n$ ($n=2,3$ or higher). Exact solutions in $\mathbb{R}^2$ and $\mathbb{R}^3$ in…
Machine Learning (ML) methods and tools have gained great success in many data, signal, image and video processing tasks, such as classification, clustering, object detection, semantic segmentation, language processing, Human-Machine…
Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve…
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…