Upwind, No More: Flexible Traveltime Solutions Using Physics-Informed Neural Networks

In recent engineering applications using deep learning, physics-informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential equations (PDE) for the loss function. Most PINNs are implemented using automatic differentiation (AD) for training the PDE loss functions. A lesser well-known study is the use of finite difference method (FDM) as an alternative. Unlike an AD based PINN, an immediate benefit of using a FDM based PINN is low implementation cost. In this paper, we propose the use of finite difference method for estimating the PDE loss functions in PINN. Our work is inspired by computational analysis in electromagnetic systems that traditionally solve Laplace’s equation using successive over-relaxation. In the case of Laplace’s equation, our PINN approach can be seen as taking the Laplacian filter response of the neural network output as the loss function. Thus, the implementation of PINN can be very simple. In our experiments, we tested PINN on Laplace’s equation and Burger’s equation. We showed that using FDM, PINN consistently outperforms non-PINN based deep learning. When comparing to AD based PINNs, we showed that our method is faster to compute as well as on par in terms of error reduction.

CAN-PINN: A fast physics-informed neural network based on coupled-automatic–numerical differentiation method

In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. PINNs are one of the deep learning-based techniques. Based on the training data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. Using automatic differentiation, the PDEs information could be encoded into NNs and a loss function. To deal with the surface differential operators in the loss function, we combine the extrinsic approach with PINNs and then express that loss function in extrinsic form. Subsequently, the loss function could be minimized extrinsically with respect to the NN parameters. Numerical results demonstrate that the extrinsic approach based on PINNs for surface problems has good accuracy and higher efficiency compared with the embedding approach based on PINNs. In addition, the strong nonlinear mapping ability of NNs makes this approach robust in solving time-dependent nonlinear problems on more complex surfaces.

Simulating convective heat transfer using traditional numerical methods requires explicit definition of thermal boundary conditions on all boundaries of the domain, which is almost impossible to fulfill in real applications. Here, we address this ill-posed problem using machine learning techniques by assuming that we have some extra measurements of the temperature at a few locations in the domain, not necessarily located on the boundaries with the unknown thermal boundary condition. In particular, we employ physics-informed neural networks (PINNs) to represent the velocity and temperature fields while simultaneously enforce the Navier-Stokes and energy equations at random points in the domain. In PINNs, all differential operators are computed using automatic differentiation, hence avoiding discretization in either space or time. The loss function is composed of multiple terms, including the mismatch in the velocity and temperature data, the boundary and initial conditions, as well as the residuals of the Navier-Stokes and energy equations. Here, we develop a data-driven strategy based on PINNs to infer the temperature field in the prototypical problem of convective heat transfer in flow past a cylinder. We assume that we have just a couple of temperature measurements on the cylinder surface and a couple more temperature measurements in the wake region, but the thermal boundary condition on the cylinder surface is totally unknown. Upon training the PINN, we can discover the unknown boundary condition while simultaneously infer the temperature field everywhere in the domain with less than 5% error in the Nusselt number prediction. In order to assess the performance of PINN, we carried out a high fidelity simulation of the same heat transfer problem (with known thermal boundary conditions) by using the high-order spectral/hp-element method (SEM), and quantitatively evaluated the accuracy of PINN’s prediction with respect to SEM. We also propose a method to adaptively select the location of sensors in order to minimize the number of required temperature measurements while increasing the accuracy of the inference in heat transfer.

Recently, there has been a growing interest in the development of intelligent labelfree structural damage identification methods that utilize physics-informed neural networks (PINNs). However, penalizing the governing equation of training data is computationally time-consuming since the existence of high-order partial derivatives. To address this issue, a damage identification method for isotropic and homogeneous thin plates is proposed in this paper that utilizes transfer learning physics-informed neural networks (TL-PINNs). TL-PINNs are efficient PINNs that solve inverse problems by leveraging transfer learning. Transfer learning is a machine learning technique that leverages knowledge from a source task to enhance performance on a related but different target task. It involves reusing a source model trained on a source task and then fine-tuning it to a target model with a target task. In the proposed method, the source model is trained to minimize the mismatch between training data and its predictions. Then, it is finetuned as the target model by minimizing both the mismatch between training data and its predictions as well as residuals that penalize the governing equation of isotropic and homogeneous thin plates. It is resulting in fewer iterations being required in training to penalize the governing equation than those in PINNs, which is time-consuming for highorder partial derivatives using automatic differentiation. Hence, TL-PINNs have a substantial reduction in computational time compared to PINNs for damage identification. A trained TL-PINN from a measured flexural guided wavefield is referred to as a pseudopristine model since it can generate a wavefield that approximates that governed by an isotropic and homogeneous thin plate. This unique functionality arises from penalizing the governing equation in the target model and the fact that the governing equation does not consider the existence of the damage. Any local anomalies in the measured wavefield can be isolated by comparing them with the wavefield generated by the pseudo-pristine model and then intensified using the Teager energy operator. An accumulative damage index is formulated, and the damage can be identified within neighborhoods with high index values. The effectiveness of the proposed method is demonstrated through a numerical investigation. A parameter study is also conducted to investigate the robustness of TL-PINNs with different hyper-parameters.

Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

Glucocorticoid-induced hypertension affects over 30% of treated patients, yet its underlying mechanisms remain unclear, particularly how glucocorticoids regulate renin within the renin-angiotensin-aldosterone system (RAAS). Modeling these dynamics is difficult because only four dose-response measurements are available at a single 24-h timepoint (36 observations total), while the system depends on roughly eleven biochemical parameters spanning minutes-long receptor interactions to days-long protein secretion. Classical parameter estimation becomes unreliable in this extremely underdetermined setting, and purely data-driven methods offer limited biological interpretability. In this paper, we introduce a physics-informed neural network (PINN) framework that integrates ELISA measurements from As4.1 juxtaglomerular cells, ordinary differential equations describing glucocorticoid receptor signaling and renin transcription, and automatic differentiation to enforce mechanistic constraints. By systematically tuning synthetic-data weights (SW in {0.2, 0.3, 0.5}), we identify an intermediate value of SW = 0.3 that provides the best overall balance between predictive accuracy, accepted ensemble size, and biologically plausible parameter estimates among the tested configurations. The framework uses adaptive constraint scheduling with a plateau ramp to reduce premature convergence and introduces calibrated plausibility thresholds reflecting experimental noise. The accepted PINN ensemble (n = 5, 50% success rate) achieved R2 = 0.803, compared with 0.759 for the SW = 0.5 baseline and −0.220 for the ODE-only baseline, with RMSE = 0.024. Key learned parameters (IC50 = 2.925 ± 0.012 mg/dL, Hill = 1.950 ± 0.009) are biologically plausible within the model assumptions, and the best single accepted model attained R2 = 0.891. Information criteria favored the PINN over the ODE model, with improvements of approximately 77× (AIC) and 5.9× (BIC). Despite training on a single timepoint, the PINN also infers full 48-h trajectories and reproduces non-monotonic dose-response behavior. This work presents, to our knowledge, the first PINN framework for glucocorticoid-mediated renin regulation and should be interpreted as a proof-of-concept approach for integrating sparse biomedical data with mechanistic constraints. The inferred parameters and temporal dynamics are best viewed as model-dependent, hypothesis-generating estimates rather than validated biological quantities.

Partial differential equations (PDEs) are core tools for characterizing physical laws and are widely used in fluid mechanics, power systems, additive manufacturing, and other fields. However, traditional numerical methods are limited by mesh partitioning, leading to a trade-off between accuracy and efficiency in complex geometric domains and high-dimensional problems. While Physically Informed Neural Networks (PINNs) achieve meshless PDE solutions by embedding prior physical knowledge, they still face challenges such as insufficient accuracy, low training efficiency, and poor stability. This study aims to address the core bottlenecks of PINN in solving PDEs by proposing a systematic optimization strategy to improve its numerical solution accuracy, efficiency, and stability. Methodologically, firstly, a composite loss function containing PDE residuals and boundary initial condition constraints is constructed based on automatic differentiation techniques to ensure that the network satisfies physical laws. Secondly, the PINN training system is optimized, including using Neural Architecture Search (NAS-PINN) to automatically match the optimal network structure, designing an adaptive data sampling strategy to improve data utilization, and introducing parallel computing and hardware acceleration techniques to reduce training time. Finally, the effectiveness of the strategy is verified through classical PDEs and engineering problems. The results show that the optimized PINN achieves high-precision numerical approximation in classical PDE solutions and can efficiently solve forward design and backward parameter inversion problems in engineering scenarios. Compared with traditional methods, it exhibits stronger adaptability in complex geometric domains and high-dimensional problems, while improving training efficiency by more than 30% and significantly enhancing stability. This research not only enriches the optimization theory of PINN and provides an efficient new path for PDE numerical solutions, but also promotes the practical application of PINN in engineering fields such as fluid mechanics, additive manufacturing, and power systems, possessing significant theoretical value and application significance.

Physics-Informed Neural Networks (PINNs) embed governing differential equations into training, enabling solutions of ODEs and PDEs. This review consolidates theoretical foundations (expressive capacity, automatic differentiation) and core methods (loss design, constraint enforcement, sampling, optimization), while surveying applications from baseline PINNs to advanced variants such as multi-physics coupling, domain decomposition, frequency-enhanced representations, and operator-learning hybrids. Comparative synthesis links architectural and training choices to equation type, data conditions, and computational budgets. A unified benchmarking framework is proposed with standard PDE tasks, accuracy metrics, collocation budgets, and transparent reporting for fair comparison with classical solvers. Evidence positions PINNs as complementary to traditional methods-effective for inverse problems, data assimilation, irregular domains, and parametric inference-while challenges remain in scalability, spectral bias, constraint enforcement, and reproducibility. The review offers a coherent synthesis with actionable guidance for scientific and engineering applications.

Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative

Summary Traveltime (eikonal) tomography is a widely used tool for estimating the Earth’s velocity structures. Though in most situations the method provides a good initial model for subsequent velocity model building applications, it has a fundamental weakness in handling velocity reversals. To overcome this limitation, several recent works incorporate additional information from the recorded data (not only the most energetic waves) to invert for the low velocity layers. Most of these works, however, require heavy computational cost as they require either wavefield simulations or additional machine learning training prior to the inversion process. Thus, we promote a new physics-informed neural networks (PINNs) framework as a replacement to classical eikonal tomography. We enforce the model (well information) and data (traveltime picks) as hard constraints in the eikonal equation. Our preliminary results show the method performs well on 2-D complex near-surface models.

In this paper, we present the adaptive physics-informed neural networks (PINNs) for resolving three dimensional (3D) dynamic thermo-mechanical coupling problems in large-size-ratio functionally graded materials (FGMs). The physical laws described by coupled governing equations and the constraints imposed by the initial and boundary conditions are leveraged to form the loss function of PINNs by means of the automatic differentiation algorithm, and an adaptive loss balancing scheme is introduced to improve the performance of PINNs. The adaptive PINNs are meshfree and trained on batches of randomly sampled collocation points, which is the key feature and superiority of the approach, since mesh-based methods will encounter difficulties in solving problems with large size ratios. The developed methodology is tested for several 3D thermo-mechanical coupling problems in large-size-ratio FGMs, and the numerical results demonstrate that the adaptive PINNs are effective and reliable for dealing with coupled problems in coating structures with large size ratios up to 109, as well as complex large-size-ratio geometries such as the electrostatic comb, the airplane and the submarine.

Efficient physics-informed neural networks using hash encoding

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving acoustic seismic wave equations in geophysical modeling. However, they often encounter challenges such as spectral bias, gradient instability, and slow convergence when applied to complex media, high-frequency wavefields, and singular source terms. To address these limitations, we propose Adaptive SIREN-PINNs, an enhanced framework that integrates learnable amplitude and frequency modulation parameters into the Sinusoidal Representation Networks (SIRENs), which employs periodic activation functions to naturally represent continuous signals and their derivatives, along with a principled weight initialization strategy. This enhancement improves the model’s frequency-aware representation capability, enabling it to better capture both low- and high-frequency components while enhancing gradient propagation stability during training. Furthermore, to mitigate the singularity associated with point sources, we adopt a Gaussian approximation of the Dirac function and reformulate the wave equation in terms of the scattered wavefield, following established regularization practices in seismic wave modeling. Additionally, we conduct a spectral analysis of the Neural Tangent Kernel (NTK) to gain insights into the model’s learning dynamics across different frequency components during training. Numerical experiments conducted on isotropic homogeneous medium, the Marmousi model, and anisotropic TTI/VTI medium demonstrate that the proposed method significantly outperforms conventional PINNs and baseline SIREN-PINNs in terms of wavefield reconstruction accuracy and convergence efficiency. The results validate the effectiveness of the adaptive SIREN mechanism in improving both the expressive power and training stability of PINNs, offering a mesh-free and solver-free alternative to traditional numerical methods.

Accurate forecasting of the lifetime and degradation mechanisms of lithium-ion batteries is crucial for their optimization, management, and safety while preventing latent failures. However, the typical state estimations are challenging due to complex and dynamic cell parameters and wide variations in usage conditions. Physics-based models need a tradeoff between accuracy and complexity due to vast parameter requirements, while machine-learning models require large training datasets and may fail when generalized to unseen scenarios. To address this issue, this paper aims to integrate the physics-based battery model and the machine learning model to leverage their respective strengths. This is achieved by applying the deep learning framework called physics-informed neural networks (PINN) to electrochemical battery modeling. The state of charge and state of health of lithium-ion cells are predicted by integrating the partial differential equation of Fick’s law of diffusion from a single particle model into the neural network training process. The results indicate that PINN can estimate the state of charge with a root mean square error in the range of 0.014% to 0.2%, while the state of health has a range of 1.1% to 2.3%, even with limited training data. Compared to conventional approaches, PINN is less complex while still incorporating the laws of physics into the training process, resulting in adequate predictions, even for unseen situations.