Physics-informed neural networks solve partial differential equations by training neural networks. Since this method approximates infinite-dimensional PDE solutions with finite collocation points, minimizing discretization errors by selecting suitable points is essential for accelerating the learning process. Inspired by number theoretic methods for numerical analysis, we introduce good lattice training and periodization tricks, which ensure the conditions required by the theory. Our experiments demonstrate that GLT requires 2-7 times fewer collocation points, resulting in lower computational cost, while achieving competitive performance compared to typical sampling methods.

The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning of solution operators of partial differential equations (PDEs). However, these works still struggle to learn dynamics that obeys the laws of physics. This paper proposes Energy-consistent Neural Operators (ENOs), a general framework for learning solution operators of PDEs that follows the energy conservation or dissipation law from observed solution trajectories. We introduce a novel penalty function inspired by the energy-based theory of physics for training, in which the functional derivative is calculated making full use of automatic differentiation, allowing one to bias the outputs of the DNN-based solution operators to obey appropriate energetic behavior without explicit PDEs. Experiments on multiple systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.

Deep learning has achieved great success in modeling dynamical systems, providing data-driven simulators to predict complex phenomena, even without known governing equations. However, existing models have two major limitations: their narrow focus on mechanical systems and their tendency to treat systems as monolithic. These limitations reduce their applicability to dynamical systems in other domains, such as electrical and hydraulic systems, and to coupled systems. To address these limitations, we propose Poisson-Dirac Neural Networks (PoDiNNs), a novel framework based on the Dirac structure that unifies the port-Hamiltonian and Poisson formulations from geometric mechanics. This framework enables a unified representation of various dynamical systems across multiple domains as well as their interactions and degeneracies arising from couplings. Our experiments demonstrate that PoDiNNs offer improved accuracy and interpretability in modeling unknown coupled dynamical systems from data.

Physical phenomena in the real world are often described by energy-based modeling theories, such as Hamiltonian mechanics or the Landau theory. It is known that physical phenomena based on these theories have an energy conservation law or a dissipation law. Therefore, in the simulations of such physical phenomena, numerical methods that preserve the energy-conservation or dissipation laws are desirable. However, because various energy-behavior-preserving numerical methods have been proposed, it is difficult to discover the best one. In this study, we propose a method for learning highly accurate energy-behavior-preserving integrators from data. Numerical results show that our approach certainly learns energy-behavior-preserving numerical methods that are more accurate than existing numerical methods for various differential equations, including chaotic Hamiltonian systems, dissipative systems, and a nonlinear partial differential equation. We also provide universal approximation theorems for the proposed approach.

Modeling and simulating physical phenomena, especially those governed by partial differential equations (PDEs), pose significant challenges in computational physics and scientific machine learning. While neural network approaches have made strides in learning continuous-time dynamics, they have struggled with discrete-time scenarios and often fail to adhere to fundamental laws of physics, such as the conservation of energy and mass. This study addresses this gap by introducing a novel deep energy-based discrete-time model. In the real world, energy-based modeling theories like Hamiltonian mechanics and the Landau theory are pivotal, as they support various laws of physics. By integrating differential geometric structures into neural networks as coefficient matrices, our model successfully simulates the conservation and dissipation laws of energy and mass. Furthermore, we propose an automatic discrete differentiation algorithm, which enables neural networks to utilize the discrete gradient method, ensuring adherence to these laws in discrete-time settings. This capability also facilitates the identification of such laws directly from data by learning matrices that represent geometric structures. These advantages are verified using simulation results of physical phenomena, namely the 1- and 2-D Korteweg-de Vries (KdV) equation and the Cahn-Hilliard equation.

The combination of neural networks and numerical integration can provide highly accurate models of continuous-time dynamical systems and probabilistic distributions. However, if a neural network is used n times during numerical integration, the whole computation graph can be considered as a network n times deeper than the original. The backpropagation algorithm consumes memory in proportion to the number of uses times of the network size, causing practical difficulties. This is true even if a checkpointing scheme divides the computation graph into subgraphs. Alternatively, the adjoint method obtains a gradient by a numerical integration backward in time; although this method consumes memory only for single-network use, the computational cost of suppressing numerical errors is high. The symplectic adjoint method proposed in this study, an adjoint method solved by a symplectic integrator, obtains the exact gradient (up to rounding error) with memory proportional to the number of uses plus the network size. The theoretical analysis shows that it consumes much less memory than the naive backpropagation algorithm and checkpointing schemes. The experiments verify the theory, and they also demonstrate that the symplectic adjoint method is faster than the adjoint method and is more robust to rounding errors.

Many physical phenomena are described by Hamiltonian mechanics using an energy function (Hamiltonian). Recently, the Hamiltonian neural network, which approximates the Hamiltonian by a neural network, and its extensions have attracted much attention. This is a very powerful method, but theoretical studies are limited. In this study, by combining the statistical learning theory and KAM theory, we provide a theoretical analysis of the behavior of Hamiltonian neural networks when the learning error is not completely zero. A Hamiltonian neural network with non-zero errors can be considered as a perturbation from the true dynamics, and the perturbation theory of the Hamilton equation is widely known as KAM theory. To apply KAM theory, we provide a generalization error bound for Hamiltonian neural networks by deriving an estimate of the covering number of the gradient of the multi-layer perceptron, which is the key ingredient of the model. This error bound gives a sup-norm bound on the Hamiltonian that is required in the application of KAM theory.

Localization using low-power wireless devices is one of the promising methods even in outdoor environments because the Global Navigation Satellite System (GNSS) devices generally consume much power to result in frequent battery replacements. Received Signal Strength Indicator (RSSI) is often utilized in such methods due to availability in most wireless devices. Even in outdoor open space, however, various factors such as path-loss characteristics and antenna directivity possibly cause difficulty in RSSI-based localization. This paper investigates the error factors of Deep Neural Network (DNN)-based fingerprinting localization in terms of antenna directivity. Most existing studies have analyzed the error factor in localization algorithms or methods to improve accuracy. On the other hand, the error factors on the environments had not been paid attention yet at all. In this paper, we investigate distance errors of the DNN-based fingerprinting method in terms of antenna directivity as one of the environmental factors using virtual space reproduced in a computer. Through experiments where an actual Bluetooth Low Energy (BLE) tag system was emulated, we found that the tag antenna directivity could be the factor to worsen the localization accuracy while receiver antenna directivity to better.

In recent years, substantial research on the methods for learning Hamiltonian equations has been conducted. Although these approaches are very promising, the commonly used representation of the Hamilton equation uses the generalized momenta, which are generally unknown. Therefore, the training data must be represented in this unknown coordinate system, and this causes difficulty in applying the model to real data. Meanwhile, Hamiltonian equations also have a coordinate-free expression that is expressed by using the symplectic 2-form. In this study, we propose a model that learns the symplectic form from data using neural networks, thereby providing a method for learning Hamiltonian equations from data represented in general coordinate systems, which are not limited to the generalized coordinates and the generalized momenta. Consequently, the proposed method is capable not only of modeling target equations of both Hamiltonian and Lagrangian formalisms but also of extracting unknown Hamiltonian structures hidden in the data. For example, many polynomial ordinary differential equations such as the Lotka-Volterra equation are known to admit non-trivial Hamiltonian structures, and our numerical experiments show that such structures can be certainly learned from data. Technically, each symplectic 2-form is associated with a skew-symmetric matrix, but not all skew-symmetric matrices define the symplectic 2-form. In the proposed method, using the fact that symplectic 2-forms are derived as the exterior derivative of certain differential 1-forms, we model the differential 1-form by neural networks, thereby improving the efficiency of learning.

A neural network model of a differential equation, namely neural ODE, has enabled the learning of continuous-time dynamical systems and probabilistic distributions with high accuracy. The neural ODE uses the same network repeatedly during a numerical integration. The memory consumption of the backpropagation algorithm is proportional to the number of uses times the network size. This is true even if a checkpointing scheme divides the computation graph into sub-graphs. Otherwise, the adjoint method obtains a gradient by a numerical integration backward in time. Although this method consumes memory only for a single network use, it requires high computational cost to suppress numerical errors. This study proposes the symplectic adjoint method, which is an adjoint method solved by a symplectic integrator. The symplectic adjoint method obtains the exact gradient (up to rounding error) with memory proportional to the number of uses plus the network size. The experimental results demonstrate that the symplectic adjoint method consumes much less memory than the naive backpropagation algorithm and checkpointing schemes, performs faster than the adjoint method, and is more robust to rounding errors.

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 mimicking of the energy conservation law by learning the underlying continuous-time differential equations. However, this may not be possible in discrete time, which is often the case in practical learning and computation. Moreover, other physical laws have been overlooked in the previous neural network models. In this study, we propose a deep energy-based physical model that admits a specific differential geometric structure. From this structure, the conservation or dissipation law of energy and the mass conservation law follow naturally. To ensure the energetic behavior in discrete time, we also propose an automatic discrete differentiation algorithm that enables neural networks to employ the discrete gradient method.