Physics-informed Neural Networks Research Articles

Physics informed neural networks for an inverse problem in peridynamic models

AbstractDeep learning is a powerful tool for solving data driven differential problems and has come out to have successful applications in solving direct and inverse problems described by PDEs, even in presence of integral terms. In this paper, we propose to apply radial basis functions (RBFs) as activation functions in suitably designed Physics Informed Neural Networks (PINNs) to solve the inverse problem of computing the perydinamic kernel in the nonlocal formulation of classical wave equation, resulting in what we call RBF-iPINN. We show that the selection of an RBF is necessary to achieve meaningful solutions, that agree with the physical expectations carried by the data. We support our results with numerical examples and experiments, comparing the solution obtained with the proposed RBF-iPINN to the exact solutions.

Temporal and spatial flow field reconstruction from low-resolution PIV data and pressure probes using physics-informed neural networks

In this paper, we present an innovative approach using physics-informed neural networks to reconstruct high-frequency, full-field flows, including the pressure field, by integrating sparse, noisy, low-temporal-resolution particle image velocimetry (PIV) data with high-temporal-resolution pressure probe data. This method effectively leverages the spatial richness of PIV data and the temporal abundance of pressure probe data, offering a complementary spatial and temporal data fusion. The incorporation of physical laws via equation constraints enables the neural network to accurately learn and predict complex fluid dynamics, achieving comprehensive flow field reconstructions. Rigorous testing across various flow types—periodic, non-periodic, and complex—against critical parameters like sampling interval (SI), number of grid points per snapshot (NGPIS), and number of pressure probes (NPP) has demonstrated remarkable accuracy. The results show reconstruction errors for velocity components (u, v) and pressure (p) below 5% with sufficient data, and around 10% for v and p, and below 5% for u in data-limited scenarios. A case study with SI = 30, NGPIS = 2000, NPP = 5 underscores the enhanced robustness and accuracy of random sampling, especially under various noisy conditions. Thus, this approach shows significant potential for temporal and spatial reconstruction of flow fields.

Physics-driven neural networks for nonlinear micromechanics

Micromechanical problems seek full-field solutions in response to external and/or internal thermal-mechanical loads, which have been increasingly encountered in material property and process assessments. Here we present a machine learning based strategy of solving micromechanics at finite strains. Deep neural networks (DNNs) are trained by employing the elastic energy together with other physical constraints to formulate the loss function. In particular, the crossover of material points, which can be a common issue for similar DNNs when applied to compression-dominant problems, is shown to be effectively addressed by including a kinematic penalty term that forces the DNN to maintain structural integrity and stability and avoid unphysical behavior. The generality and accuracy of the proposed physics-driven neural networks (PDNNs) are demonstrated through various micromechanical problems, including single crystal anisotropy elasticity, Eshelby's inclusion problem, buckling, and elastic homogenization of polycrystals. The effect of the choice of optimizers and hyperparameters on the PDNN training are discussed and the computational efficiency is also analyzed. It is shown that this novel PDNN framework can completely remove the need of labeled training data and exhibit improved performances as compared to the conventional physics-informed neural networks (PINNs) in terms of avoiding unphysical solutions and attaining higher computational efficiency. PDNNs can thus serve as a promising tool to solve some critical challenges associated with a wide range of nonlinear micromechanical problems.

Physics-informed shape optimization using coordinate projection

The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework’s capability as a viable tool for shape optimization in complex material design tasks.

Form-finding of frame-supported non-minimal tensile membrane structures for anisotropic prestress using physics-informed neural networks

Form-finding of frame-supported non-minimal tensile membrane structures for anisotropic prestress using physics-informed neural networks

Physics-informed neural networks for friction-involved nonsmooth dynamics problems

Physics-informed neural networks for friction-involved nonsmooth dynamics problems

Advancing ultimate bond stress–slip model of UHPC structures through a novel hybrid machine learning approach

The significance of Ultimate Bond Stress-Slip (UBS-S) in reinforced Ultra-High Performance Concrete (UHPC) structures cannot be overstated, as it directly affects their load-carrying capacity, structural integrity, and long-term performance. A comprehensive analysis of the UHPC-Parallel Micro Element System (UHPC-PMES), including 144 specimens, evaluated the computational efficiency of the proposed UBS-S model. To this end, the most critical settings of steel bar diameter, concrete cover, bond length, and UHPC compressive strength (db,c,lb,fUHPC′) were directed to create this parametric research. Applying the hybrid approach of three different optimization techniques, namely Physics-Informed Neural Networks (PINN), Genetic Algorithms (GA), and Multiple Linear Regression (MLR), this study predicted the UBS-S at the interface of UHPC and steel bars. It formulated the hyper-parameters effect values (a,m,β,G). The presented research used these algorithms to solve an inverse problem in structural engineering. Comparing the results obtained from PINN, GA, and MLR demonstrated that machine learning techniques and the proposed PMES model could effectively and accurately investigate the ultimate bond stress-slip for reinforced UHPC structures.

Robust Variational Physics-Informed Neural Networks

We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov–Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN’s loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection–diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp.

Learning scattering waves via coupling physics-informed neural networks and their convergence analysis

Time harmonic acoustic wave scattering problems modeled by the unbounded Helmholtz equations are widely used in medical and military fields. A fast and flexible solver for the Helmholtz equation is expected in engineering applications. Fortunately, data-driven neural network algorithms show extraordinary promise in scientific computing and artificial intelligence. We solve the Helmholtz equation by decomposing the equation into a coupling equation system that explicitly includes the real and imaginary parts of the solution. The real and imaginary parts are approximated by the different neural networks, respectively. The unbounded domain is reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator. Thus, a coupling physics-informed neural network (cPINN) is constructed. On the other hand, in exploring the convergence analysis for PINNs, we obtain the convergence results for cPINNs through the universal approximation and Sobolev space theory instead of using the neural tangent kernel theory. We prove that the cPINNs converge to the solution of the Helmholtz equation in H01 space as the number of neurons approaches infinity. Finally, we propose some numerical results.

DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus

Modified Benney-Luke equation (mBL equation) is a three-dimensional temporal-spatial equation with complex structures, that is a high-dimensional partial differential equation (PDE), it is also a new equation of the physical ocean field, and its solution is important for studying the internal wave-wave interaction of inclined seafloor. For conventional PDE solvers such as the pseudo-spectral method, it is difficult to solve mBL equation with both accuracy and speed. Physics-informed neural network (PINN) incorporates physical prior knowledge in deep neural networks, which can solve PDE with relative accuracy and speed. However, PINN is only suitable for solving low-dimensional PDE with simple structures, and not suitable for solving high-dimensional PDE with complex structures. This is mainly because high-dimensional PDEs usually have complex structures and high-order derivatives and are likely to be high-dimensional non-convex functions, and the high-dimensional non-convex optimization problem is an NP-hard problem, resulting in the PINN easily falling into inaccurate local optimal solutions when solving high-dimensional PDEs. Therefore, we improve the PINN for the characteristics of mBL equation and propose “DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus” to solve mBL equation with large amounts of data. DF-ParPINN consists of three modules: temporal-spatial division module of overall velocity potential field, data rational selection module of multiple time slices, and parallel computation module of high-velocity fields and low-velocity fields. The experimental results show that the solution time of DF-ParPINN is no more than 0.5s, and its accuracy is much higher than that of PINN, PIRNN, cPINN, and DeepONet. Moreover, the relative error of DF-ParPINN after deep training 1000000 epochs can be reduced to less than 0.1. The validity of DF-ParPINN proves that the improved PINN also can solve high dimensional PDE with complex structures and large amounts of data quickly and accurately, which is of great significance to the deep learning of the physical ocean field.

Simulation-free reliability analysis with importance sampling-based adaptive training physics-informed neural networks: Method and application to chloride penetration

Surrogate model-based reliability analysis aims at building a cheap-to-evaluate mathematical model as a substitute for the original performance function to enhance computational efficiency. Data-driven surrogate models have been popularly studied from a perspective of active learning. On the other hand, Physics-informed Neural Networks, called PINNs, have recently gained much attention as a physics-informed surrogate model to directly solve partial differential equations. Building on the capability of avoiding the simulation of traditional numerical solvers such as the finite element analysis, the PINN-based reliability analysis can achieve highly efficient simulation-free uncertainty quantification. This paper focuses on the development of the PINN-based reliability analysis method and its application in practical engineering applications. Reliability analysis with Importance Sampling-based Adaptive Training Physics-informed Neural Networks (IAT-PINN-RA) is proposed in this work. Compared with the existing PINN-based reliability analysis methods, IAT-PINN-RA introduces a pre-training stage for the establishment of the importance sampling distribution, and therefore achieves better performance when handling rare events. The modeling and reliability analysis of chloride penetration, which can pose serious challenges to the durability of concrete structures, are investigated. A practical example demonstrates the feasibility of using PINNs to model this physical phenomenon and the performance of the proposed method to achieve accurate and efficient reliability analysis results.

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem’s Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method.

WPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws

wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws

Inverse physics–informed neural networks for digital twin–based bearing fault diagnosis under imbalanced samples

In actual engineering, insufficient bearing samples for each fault category presents a substantial obstacle to the intelligent fault diagnosis of rolling bearings. To address sample imbalance, this work explores a novel bearing fault data–generation approach based on digital twin technique. First, an inverse physics–informed neural network (PINN) is built to recognize dynamic model parameters by embedding a bearing dynamic model into a neural network. In this network, a boundary loss is designed to quickly determine the approximate ranges of parameters that can accelerate network convergence, and a true value loss is constructed for the assessment of spectral discrepancy between simulated and actual data. Then, using an inverse PINN, a bearing fault dynamic model, and real vibration data, we propose a digital twin–based fault data–generation method for producing high-quality bearing fault samples under multiple working conditions and fault modes. Finally, the developed approach is applied to generate bearing fault vibration samples under a specific working condition. The samples are used for training the diagnostic network, thus solving the issue of sample imbalance. The comparison results of several experiments suggest that the developed data-generation method effectively improves the precision of cross-working-condition bearing fault diagnosis and surpasses multiple state-of-the-art methods.

Gradient-annihilated PINNs for solving Riemann problems: Application to relativistic hydrodynamics

We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that forces the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss. The performance of our methodology is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with the GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative l2 error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference “ground truth”, and with the corresponding error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by the GA-PINN model is comparable to that obtained with a shock-capturing scheme, achieving a performance superior to that of the baseline PINN algorithm in general. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved variables, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN methodology discussed in this paper could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions.

AI-Aristotle: A physics-informed framework for systems biology gray-box identification.

Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework-named AI-Aristotle-combines the eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility, and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. To test the performance of AI-Aristotle, we use sparse synthetic data perturbed by uniformly distributed noise. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.

Combined analysis of thermofluids and electromagnetism using physics-informed neural networks

A physics-informed neural network was developed for estimating a solution to a multi-physics problem involving electromagnetism, fluid dynamics, and heat transfer. The multi-physical phenomenon was modeled on a cylindrical conductor with electrical and magnetic field, as well as heat transfer between the conductor and the surrounding. For improved performance, the physics-informed neural network was divided into seven interconnected neural networks. Domain decomposition and variable separation maximization was achieved by optimizing each neural network and the transfer of data between them. Results generated by the proposed physics-informed neural network showed less than 2% errors when compared to those of analytical analyses and traditional numerical methods.

Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations

We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schrödinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT-symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.

A review of mechanistic learning in mathematical oncology.

Mechanistic learning refers to the synergistic combination of mechanistic mathematical modeling and data-driven machine or deep learning. This emerging field finds increasing applications in (mathematical) oncology. This review aims to capture the current state of the field and provides a perspective on how mechanistic learning may progress in the oncology domain. We highlight the synergistic potential of mechanistic learning and point out similarities and differences between purely data-driven and mechanistic approaches concerning model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Four categories of mechanistic learning (sequential, parallel, extrinsic, intrinsic) of mechanistic learning are presented with specific examples. We discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. Example applications address complex problems predominantly from the domain of oncology research such as longitudinal tumor response predictions or time-to-event modeling. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input data which are embraced in a mechanistic learning framework.

A physics-informed GAN framework based on model-free data-driven computational mechanics

Model-free data-driven computational mechanics, first proposed by Kirchdoerfer and Ortiz, replace phenomenological models with numerical simulations based on sample datasets in strain–stress space. In this study, we integrate this paradigm within physics-informed generative adversarial networks (GANs). We enhance the conventional physics-informed neural network framework by implementing the principles of data-driven computational mechanics into GANs. Specifically, the generator is informed by physical constraints, while the discriminator utilizes the closest strain–stress data to discern the authenticity of the generator’s output. This combined approach presents a new formalism to harness data-driven mechanics and deep learning to simulate and predict mechanical behaviors.