Physics-informed Neural Networks Research Articles

Research on Modeling Method of Autonomous Underwater Vehicle Based on a Physics-Informed Neural Network

Accurately modeling the system dynamics of autonomous underwater vehicles (AUVs) is imperative to facilitating the implementation of intelligent control. In this research, we introduce a physics-informed neural network (PINN) method to model the dynamics of AUVs by integrating dynamical equations with deep neural networks. This integration leverages the nonlinear expressive power of deep neural networks, alongside the robust foundation of physical prior knowledge, resulting in an AUV model proficient in long-term motion forecasting. The experimental results indicate that this method is capable of effectively extracting AUV system dynamics from datasets, exhibiting strong generalization capabilities and achieving robust long-term motion prediction. Furthermore, a model predictive control method is proposed, using the learned PINN as the predictive model to accurately track the closed-loop trajectory. This research offers novel perspectives on the dynamics modeling of AUVs and has the potential to be applied in other relevant research endeavors.

Physics-informed neural network for diffusive wave model

The diffusive wave model (DWM), a nonlinear second-order simplified form of the shallow water equation, has been widely used in hydraulic, hydrologic and irrigation engineering. Solution of the forward problem of the DWM can be utilized to predict evolution in water levels and discharge. Solution of its inverse problem allows for the identification of crucial parameters (such as Manning coefficient, rainfall intensity, etc.) based on observations. This paper applied the physics-informed neural network (PINN) with novel improvements to solve the DWM for both forward and inverse problems. In the forward problem, compared to traditional numerical methods, PINN was able to predict the evolution at any location. In the inverse problem, PINN provided a simple and efficient solution process. In order to overcome the gradient explosion in the training process caused by the characteristics of the DWM, the stop-gradient technique was adopted to train the neural network. To improve the estimation of DWM parameters, the concept of time division was developed, and a new network structure was proposed. To verify the effectiveness of PINN and its improved algorithm for DWM, seven examples were simulated. The PINN solutions for forward problems were compared with the results obtained by classical numerical methods, while the correct rainfall pattern was identified for the inverse problem.

The data-driven rogue waves of the Hirota equation by using Mix-training PINNs approach

Recently, solving differential equations by deep neural networks has attracted lots of attention, however, when the solution of differential equations has localized areas such as the rogue waves, the classical physics-informed neural networks (PINNs) cannot guarantee its prediction accuracy. In this paper, we propose two neural networks methods: mix-training physics-informed neural networks (MTPINNs) and prior information mix-training physics-informed neural networks (PMTPINNs), two deep learning models with more approximation ability based on PINNs. A series of numerical experiments show that the learning efficiency and absolute error accuracy of our proposed model can be improved significantly. The PMTPINNs model can so as to make up for the shortcomings of PINNs in this aspect. Furthermore, by testing the robustness of these two methods, for two kinds of rogue waves of the Hirota equation, it is surprising that these models also have good performance and thereby may provide more possibilities for solving mathematical physics problems better.

Distributed Physics-Informed machine learning strategies for two-phase flows

This study delves into applying Distributed Learning Machines (DLM), a subset of Physics-Informed Neural Networks (PINN), in tackling benchmark challenges within two-phase flows. Specifically, two variants of DLM, namely Distributed Physics-Informed Neural Networks (DPINN) and Transfer Physics-Informed Neural Networks (TPINN), are studied. The DLM architecture strategically divides the global domain into distinct non-overlapping sub-domains, with interconnected solutions facilitated by interface conditions embedded in the loss function. Forward and inverse benchmark problems in two-phase flows are explored: (a) bubble in a reversing vortex and (b) Bubble Rising under Buoyancy. The Volume of Fluid (VOF) method handles interface transport in both scenarios, with the inverse problem incorporating interface-position data during the training phase. The forward problem highlights the effectiveness of DPINN in capturing the interface using a simple transport equation. The distinctive contribution of this work lies in its exploration of the inverse problem, offering insights into the scalability of distributed architectures when dealing with a system of governing equations. Following the validation of an initial PINN model against Computational Fluid Dynamics (CFD) data, the study extends to DPINN and TPINN. A parametric study optimizes network hyperparameters, emphasizing the regularization of loss terms within the DPINN loss function. A self-adaptive weighting strategy based on a Gaussian probabilistic model dynamically adjusts loss weights during training to overcome challenges associated with manual parameter tuning. The evaluation of accuracy against CFD data and published results underscore the efficacy of DLMs in addressing two-phase flow problems. Additionally, the computational efficiency of distributed networks is explored compared to traditional PINNs.

Physics-informed neural networks for high-resolution weather reconstruction from sparse weather stations.

The accurate provision of weather information holds immense significance to many disciplines. One example corresponds to the field of air traffic management, in which one basis for weather detection is set upon recordings from sparse weather stations on ground. The scarcity of data and their lack of precision poses significant challenges to achieve a detailed description of the atmosphere state at a certain moment in time. In this article, we foster the use of physics-informed neural networks (PINNs), a type of machine learning (ML) architecture which embeds mathematically accurate physics models, to generate high-quality weather information subject to the regularization provided by the Navier-Stokes equations. The application of PINNs is oriented to the reconstruction of dense and precise wind and pressure fields in areas where only a few local measurements provided by weather stations are available. Our model does not only disclose and regularize such data, which are potentially corrupted by noise, but is also able to precisely compute wind and pressure in target areas. The effect of time and spatial resolution over the capability of the PINN to accurately reconstruct fluid phenomena is thoroughly discussed through a parametric study, concluding that a proper tuning of the neural network's loss function during training is of utmost importance.

Predicting transformer temperature field based on physics‐informed neural networks

AbstractThe safe operation of oil‐immersed transformers is critical to the safety and stability of the power grid. As the operating time increases, the failure rate of oil‐immersed transformers shows an increasing trend, posing serious challenges to safe operation. It is necessary to investigate the internal state of the oil‐immersed transformer to improve the digital degree and achieve digitalisation and intelligent operation and maintenance. A physics‐informed neural network (PINN) for oil‐immersed transformers was introduced to reconstruct the temperature distribution inside the transformer. According to the approach, the loss function of the network would be optimised by incorporating physical constraint loss terms including heat transfer equations, initial conditions and boundary conditions. The results show that the method proposed can be used to reconstruct and predict the temperature field of transformers in a few seconds with satisfactory accuracy. In conclusion, the PINN proposed outperforms deep neural networks in terms of accuracy, reliability and interpretability, especially in data‐poor cases.

Physics-informed neural networks for an optimal counterdiabatic quantum computation

A novel methodology that leverages physics-informed neural networks to optimize quantum circuits in systems with NQ qubits by addressing the counterdiabatic (CD) protocol is introduced. The primary purpose is to employ physics-inspired deep learning techniques for accurately modeling the time evolution of various physical observables within quantum systems. To achieve this, we integrate essential physical information into an underlying neural network to effectively tackle the problem. Specifically, the imposition of the solution to meet the principle of least action, along with the hermiticity condition on all physical observables, among others, ensuring the acquisition of appropriate CD terms based on underlying physics. This approach provides a reliable alternative to previous methodologies relying on classical numerical approximations, eliminating their inherent constraints. The proposed method offers a versatile framework for optimizing physical observables relevant to the problem, such as the scheduling function, gauge potential, temporal evolution of energy levels, among others. This methodology has been successfully applied to 2-qubit representing H2 molecule using the STO-3G basis, demonstrating the derivation of a desirable decomposition for non-adiabatic terms through a linear combination of Pauli operators. This attribute confers significant advantages for practical implementation within quantum computing algorithms.

Swirling flow field reconstruction and cooling performance analysis based on experimental observations using physics-informed neural networks

The design of thermal protection modules (such as film cooling) for combustion chambers requires a high-fidelity swirling flow field. Although numerical methods provide insights into three-dimensional mechanisms of swirling flow, their predictions of key features such as recirculation zones and swirling jets are often unsatisfactory due to inherent anisotropy and the isotropic nature of the Boussinesq hypothesis. Experimental methods, such as hot wire, laser Doppler velocimetry, and planar particle image velocimetry (PIV), offer more accurate reference data but are limited by sparse or planar observations. In this study, considering the outperformed capability of solving inverse problems, physics-informed neural network (PINN) was adopted to reconstruct the mean swirling flow field based on limited experimental observations from two-dimensional and two-component (2D2C) results. It was found that adding partial information characterizing the swirling flow, such as the swirling jet, could significantly improve the reconstruction of flow field. In addition, film cooling effectiveness was the key variable to evaluate the film cooling performance, which was relatively measurable in the scalar field. To further improve the accuracy of the reconstruction, the multi-source strategy was adopted into the neural network, where the film cooling effectiveness (FCE) of the effusion plate was imported as the scalar source. It was found that the prediction of the flow field near the target plate was improved, where the highest error reduction could reach 76.5%. Finally, through the reconstructed three-dimensional vortex distribution, it was found that swirling flow vortex structures near the swirler exit had a significant impact on cooling effectiveness, causing a non-uniform cooling distribution. This study aims to diagnose the three-dimensional swirling flow field with deep learning by leveraging limited experimental data and deepen the understanding of effusion cooling under swirling flow condition so that obtains a more accurate reference in the design of thermal protection modules.

Weak-formulated physics-informed modeling and optimization for heterogeneous digital materials.

Numerical solutions to partial differential equations (PDEs) are instrumental for material structural design where extensive data screening is needed. However, traditional numerical methods demand significant computational resources, highlighting the need for innovative optimization algorithms to streamline design exploration. Direct gradient-based optimization algorithms, while effective, rely on design initialization and require complex, problem-specific sensitivity derivations. The advent of machine learning offers a promising alternative to handling large parameter spaces. To further mitigate data dependency, researchers have developed physics-informed neural networks (PINNs) to learn directly from PDEs. However, the intrinsic continuity requirement of PINNs restricts their application in structural mechanics problems, especially for composite materials. Our work addresses this discontinuity issue by substituting the PDE residual with a weak formulation in the physics-informed training process. The proposed approach is exemplified in modeling digital materials, which are mathematical representations of complex composites that possess extreme structural discontinuity. This article also introduces an interactive process that integrates physics-informed loss with design objectives, eliminating the need for pretrained surrogate models or analytical sensitivity derivations. The results demonstrate that our approach can preserve the physical accuracy in data-free material surrogate modeling but also accelerates the direct optimization process without model pretraining.

Hypersonic inlet flow field reconstruction dominated by shock wave and boundary layer based on small sample physics-informed neural networks

High Mach number inlets face complex challenges such as shock wave/boundary layer interference, adversely impacting the aerodynamic characteristics and stable operating boundaries. Traditional computational fluid dynamics (CFD) simulations for performance and flow field analysis are slow, and data-driven deep learning technologies struggle with predicting flow fields in complex aerodynamic phenomena like shock wave/boundary layer interactions, boundary layer separation, and Mach disks. This study introduces a rapid, robust method for reconstructing hypersonic viscous inlet flow fields using inlet geometry design parameters. This method is called shock wave and boundary layer dominated two-dimensional multiphysical field reconstruction model based on multi-scale sensory field fusion residual physical information neural network (PIMSRF_ResNet), ensuring high-precision training data. Using the optimal Pareto solution set from multi-objective optimization with a small sample intelligence method, 150 sets of multi-physics fields under Ma10 conditions with varying geometric design parameters are calculated. Compared to traditional pure data-driven models, PIMSRF_ResNet's structural similarity in the test set reaches 0.9773, with a peak signal-to-noise ratio close to 30 dB and a correlation coefficient exceeding 98%. These experimental results demonstrated that the proposed method is a promising tool for real-time prediction of complex internal flow fields dominated by high Mach number shock waves and boundary layers.

Tackling the curse of dimensionality with physics-informed neural networks

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional partial differential equations (PDEs), as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerical PDEs in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs’ and PINNs’ residual into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton–Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in less than one hour for 1000 dimensions and in 12 h for 100,000 dimensions on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

A peridynamic-informed deep learning model for brittle damage prediction

Physics-informed Neural Network (PINN) has been introduced recently to predict and understand complex physical phenomena by directly incorporating feedback from governing equations. While PINNs offer remarkable capabilities, challenges arise when addressing discontinuities and heterogeneity, potentially compromising accuracy and reliability in prediction outcomes. In this study, a novel approach that combines the principles of peridynamic (PD) theory with PINN is presented to predict quasi-static damage and crack propagation in brittle materials. To achieve high prediction accuracy and convergence rate, the linearized PD governing equation is enforced in the PINN’s residual-based loss function. The proposed PD-INN is able to learn and capture complicated displacement patterns associated with different geometrical parameters, such as pre-crack position and length. The computational method presented in this study can be implemented to any PD constitutive model like Bond-Based PD or State-Based PD as well as any horizon factor. Several enhancements like cyclical annealing schedule and deformation gradient aware optimization technique are proposed to ensure the model would not get stuck in its trivial solution. The model’s performance assessment is conducted by monitoring the behavior of loss function throughout the training process. The results obtained through PD-INN demonstrate a satisfactory level of agreement in 2D and 3D when compared with the ground truth deformation from high-fidelity techniques like PD direct numerical method and Extended finite element method (X-FEM). Our results show the ability of the nonlocal PD-INN to predict damage and crack propagation accurately and efficiently.

Physics-informed generator-encoder adversarial networks with latent space matching for stochastic differential equations

We propose a new class of physics-informed neural networks, called Physics-Informed Generator-Encoder Adversarial Networks, to effectively address the challenges posed by forward, inverse, and mixed problems in stochastic differential equations (SDEs). In these scenarios, while governing equations are known, the available data consist of only a limited set of snapshots for system parameters. Our model consists of two key components: the generator and the encoder, both updated alternately by gradient descent. In contrast to previous approaches that directly match approximated solutions with real snapshots, we employ an indirect matching operating within the lower-dimensional latent feature space. This method circumvents challenges associated with high-dimensional inputs and complex data distributions in solving SDEs. Numerical experiments indicate that, compared to existing deep learning solvers, our proposed approach not only demonstrates superior accuracy but also exhibits advantages in both computational efficiency and model complexity.

Predicting positon solutions of a family of nonlinear Schrödinger equations through deep learning algorithm

We consider a hierarchy of nonlinear Schrödinger equations (NLSEs) and forecast the evolution of positon solutions using a deep learning approach called Physics Informed Neural Networks (PINN). Notably, the PINN algorithm accurately predicts positon solutions not only in the standard NLSE but also in other higher order versions, including cubic, quartic and quintic NLSEs. The PINN approach also effectively handles two coupled NLSEs and two coupled Hirota equations. In addition to the above, we report exact second-order positon solutions of the sextic NLSE and coupled generalized NLSE. These solutions are not available in the existing literature and we construct them through generalized Darboux transformation method. Further, we utilize PINNs to forecast their behaviour as well. To validate PINN's accuracy, we compare the predicted solutions with exact solutions obtained from analytical methods. The results show high fidelity and low mean squared error in the predictions generated by our PINN model.

Singular layer physics informed neural network method for plane parallel flows

We construct in this article the semi-analytic Physics Informed Neural Networks (PINNs), called singular layer PINNs (or sl-PINNs), that are suitable to predict the stiff solutions of plane-parallel flows at a small viscosity. Recalling the boundary layer analysis, we first find the corrector for the problem which describes the singular behavior of the viscous flow inside boundary layers. Then, using the components of the corrector and its curl, we build our new sl-PINN predictions for the velocity and the vorticity by either embedding the explicit expression of the corrector (or its curl) in the structure of PINNs or by training the implicit parts of the corrector (or its curl) together with the PINN predictions. Numerical experiments confirm that our new sl-PINNs produce stable and accurate predicted solutions for the plane-parallel flows at a small viscosity.

The Potential for Material Property Characterization Using Physics-Informed Neural Networks and Ultrasonic Wave Data

ABSTRACT There is a need for reliable nondestructive test methods that can collect data from structural members and analyze the results in a rapid and efficient manner. Large amounts of test data are needed to achieve such characterization, which provides additional challenges because of their heterogeneity and complexity. Advances in machine learning, in particular physics-informed neural networks (PINN), offer potential to address these problems. PINN is a particular form of artificial neural networks (ANN) and portends notable advantages over traditional measurand analysis or purely data-driven approaches. Here, we explore the potential of heterogeneous material property characterization using PINN and ultrasonic wave data. First, several types of 1-D ultrasonic wave data are numerically simulated for a spatially heterogeneous material, and then PINN is applied to predict wave velocity, defect location, and energy dissipation. Then, three different types of defects are simulated and all defects are detected using the corresponding 2-D ultrasonic wave data and PINN. The presented results demonstrate the promise of PINN to assist with heterogeneous material characterization methods.

Flow past a random array of statistically homogeneously distributed stationary Platonic polyhedrons: Data analysis, Probability maps and Deep Learning models

We perform particle-resolved direct numerical simulations (PR-DNS) of the flow past a random array of stationary Platonic polyhedrons seeded in a tri-periodic box at Reynolds number Re=1, 10, and 100 and solid volume fraction 0.05, 0.1 and 0.2. Using Platonic polyhedrons enables us to examine the effects of particle angularity on the hydrodynamic force and torque exerted on the stationary particles in a controlled manner. We report the average drag, lift, and torque coefficients as well as their respective distribution. We explore the relationship between the particle angularity, particle angular position, flow disturbances created by the neighboring particles, and the hydrodynamic force and torque exerted on each individual particle. We attempt to extend our probability map based model (MPP), our Physics Informed neural network (PINN) model previously developed for spheres to Platonic polyhedrons, and propose an additional NN architecture. We show that ignoring the angular position and actual shape of the reference and neighboring particles constitutes a reasonable first order approximation that significantly simplifies the construction of deterministic models of hydrodynamic forces and torques at low Re=1 to moderate Re=10. At high Re=100, both the MPP model and the PINN model without any additional knowledge of the particle shape and angular position fail to properly predict the force and torque fluctuations. We then design a Convolutional Neural Network (CNN) architecture that is able to maintain a satisfactory performance at Re=100 but requires additional input data in the form of a coarse-grained velocity field around each particle. Overall, predicting the torque fluctuation remains a significant challenge.

Improving the Efficiency of Training Physics-Informed Neural Networks Using Active Learning

Improving the Efficiency of Training Physics-Informed Neural Networks Using Active Learning

A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations

Abstract Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.

Physics-informed neural networks for solving flow problems modeled by the 2D Shallow Water Equations without labeled data

This paper investigates the application of physics-informed neural networks (PINNs) to solve free-surface flow problems governed by the 2D shallow water equations (SWEs). Two types of PINNs are developed and analyzed: a physics-informed fully connected neural network (PIFCN) and a physics-informed convolutional neural network (PICN). The PINNs eliminate the need for labeled data for training by employing the SWEs, initial and boundary conditions as components of the loss function to be minimized. Results from a set of idealized and real-world tests showed that the prediction accuracy and computation time (i.e., training time) of both PINNs may be less affected by the resolution of the domain discretization when compared against solutions by a Finite Volume (FV) model. Overall, the PICN shows a better trade-off between computational speed and accuracy than the PIFCN. Also, our results for the idealized problems indicated that PINNs can provide more than 5 times higher prediction accuracy than the FV model, while the FV simulation with coarse resolution (e.g., 10 m) can provide sub-centimeter accurate (RMSE) solutions at least one order of magnitude faster than the PINNs. Results from a river flood simulation showed that PINNs delivered better speed-accuracy trade-off than the FV model in terms of predicting the water depth, while FV models outperformed the PINNs for predictions of total flow discharge.