Improved Random Feature Method for Continuous Solution Reconstruction from Sparse Observation

Assimilating field data is essential to calibrate rock properties in hydrogeological applications. While such workflows are challenging for physics‐based reservoir simulators, data‐driven Artificial Neural Networks (ANN) offer a potential solution by effectively managing noisy data to infer fluid pressure diffusion p ( x , t ). Nevertheless, the sparse observation wells in hydrogeology projects make training ANN models difficult. Therefore, we employ Physics‐Informed Neural Networks (PINN) to bridge this gap between sparse spatial data. We train both ANN and PINN on a synthetic p ( x , t ) data set within a low‐permeability porous medium containing a highly permeable material. Having an identical architecture, both models integrate direct observations of the target variable, while the PINN further optimizes the diffusion equation residual. By comparing the model output as a function of the number of observation wells, we find that the PINN outperforms the ANN when spatial observations are limited to a small number of wells. Adding noise to the training data further underscores the robustness of PINN against random measurement errors or ambient noise. Furthermore, we evaluate the effectiveness of PINN in representing a high‐permeability zone without observations inside the domain. The findings suggest that accuracy is maintained even with reduced thickness of the equivalent material, although this leads to an increase in computation time. Given the sparsity of observation wells in hydrogeological applications, PINNs potentially provide a more suitable machine learning workflow than purely data‐driven ANN for reservoir modeling.

Satellite-based research on global sea surface wind is essential for understanding and monitoring the air-sea interface dynamical processes, driving the need for accurate and efficient assessment. Spaceborne scatterometers, which are among the most relevant sensors for sea surface wind observation, play a vital role in obtaining global ocean surface wind information. However, a significant challenge associated with polar-orbiting satellites lies in data gaps caused by their orbital paths, resulting in missing observations between swaths. While several satellite-derived sea surface wind products have been developed using data assimilation (DA) techniques, these existing methods are time-consuming and require large amounts of diverse data, rendering them computationally expensive. Additionally, the iterative steps of variational algorithms can only perform linear or weakly nonlinear adjustments to the governing equations, which may pose challenges given the highly non-linear nature of these equations.In this study, we leveraged physics-informed neural networks (PINNs) techniques to reconstruct large-scale sea surface wind fields with sparse scatterometer observations from different satellites, integrating observations and filling gaps. By incorporating physical constraints into the loss function, specifically the Navier-Stokes equations, we efficiently fill the data gaps and reconstruct wind fields that not only match observational data but also adhere to physical principles. Another objective of this work is to introduce the wind speed gradient and direction parallel consistent constraints into the loss function in order to enhance the detail of the reconstructed wind field and increase the accuracy of both wind speed and direction. Structurally, the PINN resembles a fully connected neural network (FCNN), offering the advantage of automatic feature extraction. Our model not only extracts valuable information from existing data but also uncovers complex patterns and correlations in data that are difficult for traditional algorithms to capture. This approach provides a novel perspective and an alternative methodology for wind field reconstruction.The results show that PINNs can reconstruct wind fields that closely resemble realistic wind patterns, capturing large-scale structures while preserving fine-scale details, thanks to the introduction of wind speed gradient and direction parallel consistent constraints. The training time for this model is about 3 hours, using only a single GPU core. This efficiency is partly due to the fact that PINNs do not rely on ensemble methods or large datasets to produce results. Unlike traditional DA methods, PINN does not depend on an initial best-guess field for assimilating observations. While we use only a small amount of scatterometer observation data, no initial field is required to complete the reconstruction. Additionally, since the PINN represents a continuous and differentiable function, it can produce outputs at any spatial or temporal point within the training domain.Recognizing their potential for forecast models and data integration, PINNs offer a promising approach for accurate sea surface wind field reconstruction and could serve as an effective alternative to current methods.

Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs

Surrogate modeling of parameterized multi-dimensional premixed combustion with physics-informed neural networks for rapid exploration of design space

Bayesian Physics Informed Neural Networks for data assimilation and spatio-temporal modelling of wildfires

PurposeCurrent methods for flow field reconstruction mainly rely on data-driven algorithms which require an immense amount of experimental or field-measured data. Physics-informed neural network (PINN), which was proposed to encode physical laws into neural networks, is a less data-demanding approach for flow field reconstruction. However, when the fluid physics is complex, it is tricky to obtain accurate solutions under the PINN framework. This study aims to propose a physics-based data-driven approach for time-averaged flow field reconstruction which can overcome the hurdles of the above methods.Design/methodology/approachA multifidelity strategy leveraging PINN and a nonlinear information fusion (NIF) algorithm is proposed. Plentiful low-fidelity data are generated from the predictions of a PINN which is constructed purely using Reynold-averaged Navier–Stokes equations, while sparse high-fidelity data are obtained by field or experimental measurements. The NIF algorithm is performed to elicit a multifidelity model, which blends the nonlinear cross-correlation information between low- and high-fidelity data.FindingsTwo experimental cases are used to verify the capability and efficacy of the proposed strategy through comparison with other widely used strategies. It is revealed that the missing flow information within the whole computational domain can be favorably recovered by the proposed multifidelity strategy with use of sparse measurement/experimental data. The elicited multifidelity model inherits the underlying physics inherent in low-fidelity PINN predictions and rectifies the low-fidelity predictions over the whole computational domain. The proposed strategy is much superior to other contrastive strategies in terms of the accuracy of reconstruction.Originality/valueIn this study, a physics-informed data-driven strategy for time-averaged flow field reconstruction is proposed which extends the applicability of the PINN framework. In addition, embedding physical laws when training the multifidelity model leads to less data demand for model development compared to purely data-driven methods for flow field reconstruction.

State estimation in minimal turbulent channel flow: A comparative study of 4DVar and PINN

Spatio-temporal physics-informed neural networks to solve boundary value problems for classical and gradient-enhanced continua

In the past few years, deep learning and its practical form of deep neural networks have been effectively used in a wide range of research areas. Researchers applied the deep learning techniques into solving Partial Differential Equations (PDEs) achieved significant results. A current approach known as Physics-Informed Neural Network (PINN), has evolved as a remarkable method to implement deep learning with the corresponding physics laws in forms of given linear or nonlinear PDEs. Using PINNs to solve PDEs can eliminate the requirements of elements grid needed in classical computational methods, such as finite difference or finite element methods. When approximating the solution of a PDE using a machine learning algorithm, it is vital to constrain the neural network to minimize the PDE residual. In this paper, a PINN architecture is presented, which employs the transient bioheat transfer equation (tBHTE) into a neural network to predict the temperature rise. The thermal model simulates the heat conduction generated from the wave propagating from ultrasound transducers. The solution of the tBHTE using PINNs utilizes a mesh-free domain while still maintaining a certain accuracy compared to conventional numerical methods. Further study of solving tBHTE with multi-elements ultrasound transducers can be generally applied to focused ultrasound treatments in order to predict the heat generation in more complex heterogeneous domains.

Solving fluid dynamics problems mainly rely on experimental methods and numerical simulation. However, in experimental methods it is difficult to simulate the physical problems in reality, and there is also a high-cost to the economy while numerical simulation methods are sensitive about meshing a complicated structure. It is also time-consuming due to the billion degrees of freedom in relevant spatial-temporal flow fields. Therefore, constructing a cost-effective model to settle fluid dynamics problems is of significant meaning. Deep learning (DL) has great abilities to handle strong nonlinearity and high dimensionality that attracts much attention for solving fluid problems. Unfortunately, the proposed surrogate models in DL are almost black-box models and lack interpretation. In this paper, the Physical Informed Neural Network (PINN) combined with Resnet blocks is proposed to solve fluid flows depending on the partial differential equations (i.e., Navier-Stokes equation) which are embedded into the loss function of the deep neural network to drive the model. In addition, the initial conditions and boundary conditions are also considered in the loss function. To validate the performance of the PINN with Resnet blocks, Burger’s equation with a discontinuous solution and Navier-Stokes (N-S) equation with continuous solution are selected. The results show that the PINN with Resnet blocks (Res-PINN) has stronger predictive ability than traditional deep learning methods. In addition, the Res-PINN can predict the whole velocity fields and pressure fields in spatial-temporal fluid flows, the magnitude of the mean square error of the fluid flow reaches to 10−5. The inverse problems of the fluid flows are also well conducted. The errors of the inverse parameters are 0.98% and 3.1% in clean data and 0.99% and 3.1% in noisy data.

Inverse problems involving nonlinear partial differential equations (PDEs) pose significant challenges due to their ill-posed nature and reliance on sparse or noisy observations. Traditional approaches often require complete knowledge of initial and boundary conditions, which may not be available in practical scenarios. Physics-informed neural networks (PINNs) have recently emerged as a powerful approach for addressing such problems by embedding physical laws into the structure of deep neural networks. In this work, we employ PINNs to solve the inverse problem for the Gardner-Kawahara equation, a high-order nonlinear dispersive PDE that modeling wave propagation in fluids and plasmas. The proposed PINNs framework accurately reconstructs unknown parameters and solution fields from limited data, even in the presence of noise. Numerical experiments conducted under varying parameter settings and noise levels demonstrate strong agreement with exact solutions, thereby highlighting the method’s accuracy, robustness, and computational efficiency. These results confirm the potential of PINNs for inverse modeling of complex nonlinear systems, even in the absence of complete initial or boundary information, and demonstrate that they outperform traditional methods in handling sparse and noisy data.

Satellite‐based research on global sea surface winds is crucial for understanding and monitoring dynamic air‐sea interface processes, driving the need for reliable assessments and forecasts. Achieving these objectives requires accurate reconstruction of wind speeds and wind directions that assimilate real‐time observations, but current methods used for such reconstructions remain computationally expensive and challenging to solve highly nonlinear equations. By selecting multiple regions in the Pacific Ocean as case studies, we explore the fusion of sparse scatterometer observations from different satellites with physics‐informed neural networks (PINNs) to reconstruct large‐scale sea surface wind fields, effectively integrating observations and filling data gaps. The result demonstrates that PINNs can efficiently reconstruct full realistic wind fields from sparse data at a low computation cost. Moreover, we propose a novel loss function that can preserve fine‐scale details while capturing the key features of the wind field. Evaluation against reference data sets, including buoy measurements, ERA5, and cross‐calibrated multiplatform (CCMP) wind speed and direction, highlights the accuracy and robustness of the proposed method. Compared to the reanalysis data of ERA5 and CCMP, the wind field reconstructed by the PINN closely aligns with the observation. These findings underscore the potential of the PINN technique as an alternative, computationally efficient approach for operational sea surface wind reconstruction.

One of the most basic nonlinear Partial Differential Equations (PDEs) to model the effects of propagation and diffusion is Burger's equation. This puts great emphasize on seeking efficient versatile methods for finding a solution to the forward and inverse problems of this equation. The focus of this paper is to introduce a method for solving the inverse problem of Burger's equation using neural networks. With recent advances in the area of deep learning, a Physics-Informed Neural Network (PINN) is a category of neural networks that proved efficient for handling PDEs. In our work, the 1D and 2D Burger's equations are simulated by applying a PINN to a set of domain points. The training process of PINNs is governed by the PDE formula, the initial conditions (ICs), the Boundary Conditions (BCs), and the loss minimization algorithm. After training the network to predict the coefficients of the nonlinear PDE, the inverse problem of the 1D and 2D Burger's equations are solved with an error as low as 0.047 and 0.2 for 1D and 2D case studies, respectively. The wave propagation model is accomplished with an approximate training loss value of 1×e-4. The utilization of PINNs for modeling Burger's equation is a mesh-free approach that competes with the commonly used numerical methods as it overcomes the curse of dimensionality. Training the PINN model to predict the propagation and diffusion effects can also be generalized to address further detailed applications of Burger's equation with complex domains. This contributes to clinical applications such as ultrasound therapeutics.

The real-time forecasting of flood dynamics is a long-standing challenge traditionally addressed through numerical solutions of the Shallow Water Equations (SWEs). Numerical solutions of realistic flow problems using numerical schemes are often hindered by high computational costs, particularly due to the need for fine spatial and temporal discretization, complex boundary conditions, and the resolution of non-linearities inherent in the governing equations. In this study, we investigate the use of Physics-Informed Neural Networks (PINNs) to solve 1D and 2D SWEs in dam-break scenarios. The proposed PINN framework incorporates the governing partial differential equations along with the initial and boundary conditions directly within the training process of the network, ensuring physically consistent solutions. We conduct a systematic comparison of the solutions of SWE using the classical numerical scheme (Lax-Wendroff) with estimates of physics informed neural networks. For 1D SWE, a neural network is trained and validated on a dam-break problem, revealing that physics-informed models produce smoother but still acceptable estimates of wave propagation compared to standard numerical results. For 2D SWE, we consider various configurations of dam geometries along with varying initial profiles for water heights. Across all scenarios, reproduce the numerical baselines, albeit with limited accuracy, while avoiding spurious oscillations and numerical artifacts. Further tuning, achieved by incorporating numerical solutions into the PINN training, improved accuracy. This proof of concept demonstrates the potential of hybridized PINNs as a mesh-free, scalable, and generalizable framework for approximating solutions to nonlinear hyperbolic systems. Our results indicate that pre-trained, physics-informed models could serve as a viable alternative for real-time flood forecasting in complex domains.

The real-time forecasting of flood dynamics is a long-standing challenge traditionally addressed through numerical solutions of the Shallow Water Equations (SWEs). Numerical solutions of realistic flow problems using numerical schemes are often hindered by high computational costs, particularly due to the need for fine spatial and temporal discretization, complex boundary conditions, and the resolution of non-linearities inherent in the governing equations. In this study, we investigate the use of Physics-Informed Neural Networks (PINNs) to solve 1D and 2D SWEs in dam-break scenarios. The proposed PINN framework incorporates the governing partial differential equations along with the initial and boundary conditions directly within the training process of the network, ensuring physically consistent solutions. We conduct a systematic comparison of the solutions of SWE using the classical numerical scheme (Lax-Wendroff) with estimates of physics informed neural networks. For 1D SWE, a neural network is trained and validated on a dam-break problem, revealing that physics-informed models produce smoother but still acceptable estimates of wave propagation compared to standard numerical results. For 2D SWE, we consider various configurations of dam geometries along with varying initial profiles for water heights. Across all scenarios, reproduce the numerical baselines, albeit with limited accuracy, while avoiding spurious oscillations and numerical artifacts. Further tuning, achieved by incorporating numerical solutions into the PINN training, improved accuracy. This proof of concept demonstrates the potential of hybridized PINNs as a mesh-free, scalable, and generalizable framework for approximating solutions to nonlinear hyperbolic systems. Our results indicate that pre-trained, physics-informed models could serve as a viable alternative for real-time flood forecasting in complex domains.