Physics‐Informed Neural Networks for Solving the Two‐Dimensional Shallow Water Equations With Terrain Topography and Rainfall Source Terms

1-D coupled surface flow and transport equations revisited via the physics-informed neural network approach

The exploration of deep learning methodologies has recently generated significant interest in the use of Physics-Informed Neural Networks (PINNs) to address complex physical problems governed by partial differential equations (PDEs). The PINN is trained using information from physical laws, including governing PDEs, boundary conditions, and initial conditions. However, achieving a well-trained PINN typically necessitates an appropriate balance between the weights of each loss function, which can considerably increase manual effort. This paper introduces a novel training approach that integrates non-dimensionalization with a multi-task learning technique, termed Automatic Weighted Loss (AWL), to autonomously achieve an optimal balance for each loss function. In the baseline PINN training for solving time-dependent PDEs, multiple weights (usually more than six) must be manually tuned for the model, whereas this method can reduce the number of scaling weights to only one. The proposed approach, referred to as the Non-dimensionalization Automatic Weighted Loss (NDAWL), is evaluated through six free-surface flow problems modelled by the Shallow Water Equations (SWEs). Furthermore, a comparative analysis is conducted between the solutions obtained using NDAWL-PINN and those from the original PINN, which relies on manual fine-tuning of loss functions. The numerical results indicate that the NDAWL-PINN method achieves comparable or superior accuracy to the original PINN, demonstrating its effectiveness in automating the balancing of loss functions.

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.

A Physics-Informed Neural Network (PINN) was developed to solve the complex Navier-Stokes (N-S) equations for multiphase flow measurement including flowrates of individual phases in the multiphase mixture, and phase ratios. The results were compared against a mathematical based multiphase flow measurement approach and validated through a controlled loop test using oil, water, and gas. The PINN embeds fundamental fluid dynamics principles into its learning process, ensuring that the neural network adheres to the governing physical laws of motion, continuity, and conservation. Unlike traditional neural networks that rely solely on data-driven learning, PINNs incorporate fluid dynamics driven differential equations, to enhance model accuracy and reliability. This study evaluates the accuracy, efficiency, and repeatability of PINNs by comparing their results against traditional mathematical-based multiphase flow measurement (MPFM) technology and controlled test cases. The developed PINN successfully solved the Navier- Stokes (N-S) equations in their closest approximation to the original form, with minimal assumptions and simplifications, to estimate multiphase flow rates, integrating fundamental fluid dynamics principles into its learning process. The results demonstrated that the PINN-based approach achieved accuracy comparable to traditional mathematical multiphase flow measurement (MPFM) technologies, while significantly improving computational efficiency. During controlled loop tests with oil, water, and gas, the PINN model consistently captured flow dynamics with high precision, validating its effectiveness under real-world conditions. Moreover, the PINN method exhibited strong generalization capabilities, adapting to various flow regimes without the need for extensive empirical calibration. Compared to conventional MPFM technologies, PINN reduced reliance on predefined correlations and offered greater adaptability to changing fluid properties. The repeatability of results confirmed the model's reliability, positioning PINNs as a promising alternative for complex multiphase flow measurement applications. These findings highlight the potential of PINNs as an innovative, physics-guided, and data-efficient solution, effectively bridging the gap between traditional physics-based methods and modern machine learning techniques. This paper presents an innovative application of Physics-Informed Neural Networks (PINNs) for multiphase flow measurement, offering a data-driven yet physics-guided alternative to traditional mathematical MPFM technologies. By embedding fundamental physical laws into the neural network training process, the proposed approach enhances accuracy, computational efficiency, and adaptability, while minimizing reliance on extensive empirical calibration. This advancement significantly improves real-time flow monitoring, equipping practicing engineers with a more reliable, data-efficient, and scalable solution for complex multiphase flow measurement in industrial applications.

Flood forecasting is usually performed by means of physically based models or conceptual models. Data-driven methods such as Artificial Neural Networks (ANN) are a promising solution as they can extract complex non-linear dependencies between inputs and outputs with robustness, generalization, and computational efficiency. However, the ANNs still rank below the classical method of numerical solvers for Saint Venant Equations in terms of accuracy. To bridge the accuracy gap, this work focuses on devising a Physics Informed Neural Network (PINN) model for spatial-temporal scale flood forecasting based on the Saint Venant Equations. The ANN architecture is extended to PINN with the implementation of the custom Partial Differential Equation (PDE) loss. This procedure allows to quickly steer the learning algorithm toward a true solution. A synthetic dataset was developed by performing numerical simulations of the Saint Venant Equations for training and evaluation of the PINN model. The results demonstrate that the PINN model performs better than the ANN Models and is suitable for water depth forecasting.

Physics-informed neural networks for high-speed flows

Physics-informed neural networks in water and wastewater systems: a critical review.

This work proposes the inverse design of bandstop Frequency Selective Surface using a Graph based Conditional Variational Autoencoder (G-CVAE) integrated with a Physics-Informed Neural Network (PINN). This inverse design involves the prediction of FSS geometry that exhibits ultra-wide stopband characteristics. Initially, the graph convolutional network precisely extracts the topological and spatial relationships within the FSS geometrical design. The features of the graph and simulation results of the FSS dataset are used to train the CVAE, which maps the FSS physical structure and its electromagnetic behavior. The trained CVAE predicts the FSS geometries with desired frequency responses, while the PINN is incorporated to ensure physical feasibility. By monitoring the average relative error values, the simulated and predicted transmission coefficients are brought closer to each other. Also, similar approach is followed to enhance the angular stability and to achieve polarization independence in both TE and TM modes. A G-CVAE-PINN is constructed and trained using various random combinations of graph attributes and simulation outcomes, achieving an average inaccuracy of 3%. Further, one of the best designs from the predicted FSS designs is chosen for experimental validation. This predicted and experimentally validated bandstop FSS exhibits wide band rejection of 20 GHz ranging from 8 GHz to 28 GHz. The fabricated design exhibits polarization independence up to 75°in both normal and oblique angles. Thus, the predicted FSS designs are ideal for radome, EMI shielding, and satellite communications, providing efficient frequency filtering for 5G and beyond 5G networks.

The interface between data-driven learning methods and classical simulation poses an interesting field offering a multitude of new applications. In this work, we build on the notion of physics-informed neural networks (PINNs) and employ them in the area of shallow-water equation (SWE) models. These models play an important role in modeling and simulating free-surface flow scenarios such as in flood-wave propagation or tsunami waves. Different formulations of the PINN residual are compared to each other and multiple optimizations are being evaluated to speed up the convergence rate. We test these with different 1-D and 2-D experiments and finally demonstrate that regarding a SWE scenario with varying bathymetry, the method is able to produce competitive results in comparison to the direct numerical simulation with a total relative $L_2$ error of $8.9e-3$.

This paper advances the use of physics-informed neural networks (PINNs) architectures to address moving interface problems via the level set method. Originally developed for other partial differential equations-based problems, we particularly leverage physics-informed deep learning with residual adaptive networks' (PirateNet) features—including causal training, sequence-to-sequence learning, random weight factorization, and Fourier feature embeddings—and tailor them to handle problems with complex interface dynamics. Numerical experiments validate this framework on benchmark problems such as Zalesak's disk rotation and time-reversed vortex flow. We demonstrate that PINNs can efficiently solve level set problems exhibiting significant interface deformation without the need for upwind numerical stabilization, as generally required by classic discretization methods, or additional mass conservation schemes. However, incorporating an Eikonal regularization term in the loss function with an appropriate weight can further enhance results in specific scenarios. Our results indicate that PINNs with the PirateNet architecture surpass conventional PINNs in accuracy, achieving state-of-the-art error rates of L2=0.14% for Zalesak's disk and L2=0.85% for the time-reversed vortex flow problem, as compared to reference solutions. Additionally, for a complex two-phase flow dam break problem coupling the level set with the Navier–Stokes equations, we propose a geometric reinitialization method embedded within the sequence-to-sequence training scheme to ensure long-term stability and accurate inference of the level set field. The proposed framework has the potential to be broadly applicable to industrial problems that involve moving interfaces, such as free-surface flows in hydraulics and maritime engineering.

Physical information-fused deep learning model ensembled with a subregion-specific sampling method for predicting flood dynamics

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

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.

The physics‐informed neural network (PINN) method has been applied to solve water hammer equations in pipeline systems due to its ability of seamlessly integrate measurement data with conservation laws, offering advantages over traditional numerical method. However, existing PINN approaches require multiple neural networks to construct composite models for complex water distribution systems (WDS). This situation treats nodal information as boundary condition or labeled data during training, leading to a weaker robustness and a high demand for data. To address these issues, a hybrid water hammer model based on eXtended Physics‐Informed Neural Networks for WDS (WDS‐XPINN) is developed in this study. Unlike the standard PINN, WDS‐XPINN incorporates the nodal mechanistic model directly into the loss function, enabling to synchronously train a unified neural network jointly through sparse augmented measurement data for pipeline system. Additionally, an adaptive weights method is introduced to improve model robustness by balancing the contributions of flowrates and pressures. The proposed WDS‐XPINN is evaluated in two case studies: a series pipeline system with different operational events and noise perturbation, as well as a topological structure with looped and branched pipe. According to the simulation results and uncertainty analysis, the WDS‐XPINN model demonstrates its excellent capacity of modeling fluid transient accurately in pipeline system, even without exact operational conditions or true pipe parameters.

Autonomous underwater vehicles (AUVs) have played a pivotal role in advancing ocean exploration and exploitation. However, traditional AUVs face limitations when executing missions at minimal or near-zero forward velocities due to the ineffectiveness of their control surfaces, considerably constraining their potential applications. To address this challenge, this paper introduces an innovative vectored thruster system based on a 3RRUR parallel manipulator tailored for micro-sized AUVs. The incorporation of a vectored thruster enhances the performance of micro-sized AUVs when operating at minimal and low forward speeds. A comprehensive exploration of the kinematics of the thrust-vectoring mechanism has been undertaken through theoretical analysis and experimental validation. The findings from theoretical analysis and experimental confirmation unequivocally affirm the feasibility of the devised thrust-vectoring mechanism. The precise control of the vector device is studied using Physics-informed Neural Network and Model Predictive Control (PINN-MPC). Through the adoption of this pioneering thrust-vectoring mechanism rooted in the 3RRUR parallel manipulator, AUVs can efficiently and effectively generate the requisite motion for thrust-vectoring propulsion, overcoming the limitations of traditional AUVs and expanding their potential applications across various domains.