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We propose a novel dual physics-informed neural network for topology optimization (DPNN-TO), which merges physics-informed neural networks (PINNs) with the traditional SIMP-based topology optimization (TO) algorithm. This approach leverages…

We present a physically-motivated topology of a deep neural network that can efficiently infer extensive parameters (such as energy, entropy, or number of particles) of arbitrarily large systems, doing so with O(N) scaling. We use a form of…

There is a growing attention given to utilizing Lagrangian and Hamiltonian mechanics with network training in order to incorporate physics into the network. Most commonly, conservative systems are modeled, in which there are no frictional…

Neural networks are complex functions of both their inputs and parameters. Much prior work in deep learning theory analyzes the distribution of network outputs at a fixed a set of inputs (e.g. a training dataset) over random initializations…

Active researches are currently being performed to incorporate the wealth of scientific knowledge into data-driven approaches (e.g., neural networks) in order to improve the latter's effectiveness. In this study, the Theory-guided Neural…

Physics-informed neural networks (PINNs) have emerged as a powerful meshless tool for topology optimization, capable of simultaneously determining optimal topologies and physical solutions. However, conventional PINNs rely on density-based…

The past few years have witnessed an increased interest in learning Hamiltonian dynamics in deep learning frameworks. As an inductive bias based on physical laws, Hamiltonian dynamics endow neural networks with accurate long-term…

Deep Neural Networks (DNNs) are generated by sequentially performing linear and non-linear processes. Using a combination of linear and non-linear procedures is critical for generating a sufficiently deep feature space. The majority of…

We present the bulk-boundary decomposition as a new framework for understanding the training dynamics of deep neural networks. Starting from the stochastic gradient descent formulation, we show that the Lagrangian can be reorganized into a…

While Hamiltonian mechanics provides a powerful inductive bias for neural networks modeling dynamical systems, Hamiltonian Neural Networks and their variants often fail to capture complex temporal dynamics spanning multiple timescales. This…

Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep…

Controlling nonlinear stochastic dynamical systems involves substantial challenges when the dynamics contain unknown and unstructured nonlinear state-dependent terms. For such complex systems, deep neural networks can serve as powerful…

Recent research works for solving partial differential equations (PDEs) with deep neural networks (DNNs) have demonstrated that spatiotemporal function approximators defined by auto-differentiation are effective for approximating nonlinear…

Physical phenomena in the real world are often described by energy-based modeling theories, such as Hamiltonian mechanics or the Landau theory, which yield various physical laws. Recent developments in neural networks have enabled the…

In many real world applications, data are characterized by a complex structure, that can be naturally encoded as a graph. In the last years, the popularity of deep learning techniques has renewed the interest in neural models able to…

Deep learning has been the engine powering many successes of data science. However, the deep neural network (DNN), as the basic model of deep learning, is often excessively over-parameterized, causing many difficulties in training,…

The incorporation of appropriate inductive bias plays a critical role in learning dynamics from data. A growing body of work has been exploring ways to enforce energy conservation in the learned dynamics by encoding Lagrangian or…

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in…

Deep neural networks (DNNs) are widely used in pattern-recognition tasks for which a human comprehensible, quantitative description of the data-generating process, e.g., in the form of equations, cannot be achieved. While doing so, DNNs…

In this paper we study a constraint-based representation of neural network architectures. We cast the learning problem in the Lagrangian framework and we investigate a simple optimization procedure that is well suited to fulfil the…