Physics-informed Neural Networks Research Articles

Learning Reduced Fluid Dynamics

Predicting the state evolution of ultra high-dimensional, time-reversible fluid dynamic systems is a crucial but computationally expensive task. Existing physics-informed neural networks either incur high inference cost or cannot preserve the time-reversible nature of the underlying dynamics system. We propose a model-based approach to identify low-dimensional, time reversible, nonlinear fluid dynamic systems. Our method utilizes the symplectic structure of reduced Eulerian fluid and use stochastic Riemann optimization to obtain a low-dimensional bases that minimize the expected trajectory-wise dimension-reduction error over a given distribution of initial conditions. We show that such minimization is well-defined since the reduced trajectories are differentiable with respect to the subspace bases over the entire Grassmannian manifold, under proper choices of timestep sizes and numerical integrators. Finally, we propose a loss function measuring the trajectory-wise discrepancy between the original and reduced models. By tensor precomputation, we show that gradient information of such loss function can be evaluated efficiently over a long trajectory without time-integrating the high-dimensional dynamic system. Through evaluations on a row of simulation benchmarks, we show that our method reduces the discrepancy by 50-90 percent over conventional reduced models and we outperform PINNs by exactly preserving the time reversibility.

Physics-Informed Representation and Learning: Control and Risk Quantification

Optimal and safety-critical control are fundamental problems for stochastic systems, and are widely considered in real-world scenarios such as robotic manipulation and autonomous driving. In this paper, we consider the problem of efficiently finding optimal and safe control for high-dimensional systems. Specifically, we propose to use dimensionality reduction techniques from a comparison theorem for stochastic differential equations together with a generalizable physics-informed neural network to estimate the optimal value function and the safety probability of the system. The proposed framework results in substantial sample efficiency improvement compared to existing methods. We further develop an autoencoder-like neural network to automatically identify the low-dimensional features in the system to enhance the ease of design for system integration. We also provide experiments and quantitative analysis to validate the efficacy of the proposed method. Source code is available at https://github.com/jacobwang925/path-integral-PINN.

Physics-informed neural network for simulating magnetic field of coaxial magnetic gear

In the process of performance analysis and structure optimization of coaxial magnetic gear, an emerging method for precise magnetic field simulation remains a focal point in engineering research. In this study, we introduce a physics-informed neural network to model the magnetic field of a magnetic gear. We employed a physics-based loss function to optimize neural network parameters to solve the magnetic field of a Maxwell-controlled magnetic gear. Additionally, we developed a joint training model that leverages the continuity of the medium interface. The feasibility of the model was confirmed by solving the magnetic field of a permanent magnet, with an error margin of less than 5%. The model exhibited excellent precision in simulating magnetic field behavior within magnetic gears. We demonstrate that adjusting model parameters enables the creation of a proxy model, which effectively addresses analogous problems. Furthermore, leveraging transfer learning substantially diminishes training time for similar tasks, resulting in a 43% reduction in training cost. Finally, we propose an enhanced physical information neural network with data-physical drive fusion and use a special Poisson's equation solution in the magnetized region as a data drive during training. The enhanced physics-informed neural network effectively solved the magnetic field of a magnetic gear, resulting in a 50% improvement in solution accuracy. This study establishes the groundwork for analyzing and optimizing magnetic gears, providing new research insight for electromagnetic practitioners.

Aircraft Structural Design and Life-Cycle Assessment through Digital Twins

Numerical modeling tools are essential in aircraft structural design, yet they face challenges in accurately reflecting real-world behavior due to factors like material properties scatter and manufacturing-induced deviations. This article addresses the potential impact of digital twins on overcoming these limitations and enhancing model reliability through advanced updating techniques based on machine learning. Digital twins, which are virtual replicas of physical systems, offer a promising solution by integrating sensor data, operational inputs, and historical records. Machine learning techniques enable the calibration and validation of models, combining experimental inputs with simulations through continuous updating processes that refine digital twins, improving their accuracy in predicting structural behavior and performance throughout an aircraft’s life cycle. These refined models enable real-time monitoring and precise damage assessment, supporting decision making in diverse contexts. By integrating sensor data and updating techniques, digital twins contribute to improved design and maintenance operations by providing valuable insights into structural health, safety, and reliability. Ultimately, this approach leads to more efficient and safer aviation operations, demonstrating the potential of digital twins to revolutionize aircraft structural analysis and design. This article explores various advancements and methodologies applicable to structural assessment, leveraging machine learning tools. These include the utilization of physics-informed neural networks, which enable the handling of diverse uncertainties. Such approaches empower a more informed and adaptive strategy, contributing to the assurance of structural integrity and safety in aircraft structures throughout their operational life.

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.

PINNslope: seismic data interpolation and local slope estimation with physics#xD;informed neural networks

Interpolation of aliased seismic data constitutes a key step in a seismic processing workflow to obtain high-quality velocity models and seismic images. Building on the idea of describing seismic wavefields as a superposition of local plane waves, we propose to interpolate seismic data by using a physics informed neural network (PINN). In the proposed framework, two feed-forward neural networks are jointly trained using the local plane wave differential equation as well as the available data as two terms in the objective function: a primary network assisted by positional encoding is tasked with reconstructing the seismic data, while an auxiliary, smaller network estimates the associated local slopes. Results on synthetic and field data validate the effectiveness of the proposed method in handling aliased (coarsely sampled) data and data with large gaps. Our method compares favorably against a classic least-squares inversion approach regularized by the local plane-wave equation as well as a PINN-based approach with a single network and precomputed local slopes. We find that introducing a second network to estimate the local slopes while at the same time interpolating the aliased data enhances the overall reconstruction capabilities and convergence behavior of the primary network. Moreover, an additional positional encoding layer embedded as the first layer of the wavefield network confers to the network the ability to converge faster, improving the accuracy of the data term.

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.