Water movement in soil is essential for weather monitoring, prediction of natural disasters, and agricultural water management. Richardson-Richards' equation (RRE) is the characteristic partial differential equation for studying soil water movement. RRE is a non-linear PDE involving water potential, hydraulic conductivity, and volumetric water content. This equation has underlying non-linear parametric relationships called water retention curves (WRCs) and hydraulic conductivity functions (HCFs). This two-level non-linearity makes the problem of unsaturated water flow of soils challenging to solve. Physics-Informed Neural Networks (PINNs) offer a powerful paradigm to combine physics in data-driven techniques. From noisy or sparse observations of one variable (water potential), we use PINNs to learn the complete system, estimate the parameters of the underlying model, and further facilitate the prediction of infiltration and discharge. We employ training on RRE, WRC, HCF, and measured values to resolve two-level non-linearity directly instead of explicitly deriving water potential or volumetric water content-based formulations. The parameters to be estimated are made trainable with initialized values. We take water potential data from simulations and use this data to solve the inverse problem with PINN and compare estimated parameters, volumetric water content, and hydraulic conductivity with actual values. We chose different types of parametric relationships and wetting conditions to show the approach's effectiveness.

Physics-Informed Neural Networks (PINNs) have shown continuous promise in approximating partial differential equations (PDEs), although they remain constrained by the curse of dimensionality. In this paper, we propose a generalized PINN version of the classical variable separable method. To do this, we first show that, using the universal approximation theorem, a multivariate function can be approximated by the outer product of neural networks, whose inputs are separated variables. We leverage tensor decomposition forms to separate the variables in a PINN setting. By employing Canonic Polyadic (CP), Tensor-Train (TT), and Tucker decomposition forms within the PINN framework, we create robust architectures for learning multivariate functions from separate neural networks connected by outer products. Our methodology significantly enhances the performance of PINNs, as evidenced by improved results on complex high-dimensional PDEs, including the 3d Helmholtz and 5d Poisson equations, among others. This research underscores the potential of tensor decomposition-based variably separated PINNs to surpass the state-of-the-art, offering a compelling solution to the dimensionality challenge in PDE approximation.

Modeling and simulation of complex non-linear systems are essential in physics, engineering, and signal processing. Neural networks are widely regarded for such tasks due to their ability to learn complex representations from data. Training deep neural networks traditionally requires large amounts of data, which may not always be readily available for such systems. Contrarily, there is a large amount of domain knowledge in the form of mathematical models for the physics/behavior of such systems. A new class of neural networks called Physics-Informed Neural Networks (PINNs) has gained much attention recently as a paradigm for combining physics into neural networks. They have become a powerful tool for solving forward and inverse problems involving differential equations. A general framework of a PINN consists of a multi-layer perceptron that learns the solution of the partial differential equation (PDE) along with its boundary/initial conditions by minimizing a multi-objective loss function. This is formed by the sum of individual loss terms that penalize the output at different collocation points based on the differential equation and initial and boundary conditions. However, multiple loss terms arising from PDE residual and boundary conditions in PINNs pose a challenge in optimizing the overall loss function. This often leads to training failures and inaccurate results. We propose advanced gradient statistics-based weighting schemes for PINNs to address this challenge. These schemes utilize backpropagated gradient statistics of individual loss terms to appropriately scale and assign weights to each term, ensuring balanced training and meaningful solutions. In addition to the existing gradient statistics-based weighting schemes, we introduce kurtosis–standard deviation-based and combined mean and standard deviation-based schemes for approximating solutions of PDEs using PINNs. We provide a qualitative and quantitative comparison of these weighting schemes on 2D Poisson’s and Klein–Gordon’s equations, highlighting their effectiveness in improving PINN performance.

Computational Aeroacoustics (CAA) is a critical domain within computational fluid dynamics (CFD) that focuses on understanding and predicting sound generation in aerodynamic systems. Accurate estimation of Lighthill sources, which play a pivotal role in deciphering acoustic phenomena, remains a challenging task, especially when confronted with noisy and missing flow data. This study explores the potential of Physics Informed Neural Networks (PINNs) to address these challenges and capture the complex flow dynamics inherent in CAA. The integration of PINNs in CFD has gained significant attention in recent years. PINNs blend deep learning techniques with fundamental physical principles, enabling accurate predictions and enhanced modeling capabilities. Their versatility has been demonstrated across various CFD applications, ranging from turbulence modeling to flow control optimization and mesh generation. However, their potential in the field of CAA, specifically in estimating Lighthill sources from flow data, remains largely unexplored. To investigate the effectiveness of PINNs in the context of CAA, we conduct a series of experiments using high-fidelity flow data obtained from common flow configurations, such as flow around a cylinder. Leveraging this data, we create three distinct datasets that represent different data imperfections. The first dataset involves the deliberate removal of certain data points, the second dataset incorporates the addition of random noise, and the third dataset combines both missing data and noise. By incorporating the governing Navier-Stokes equations, we train the PINNs using these three datasets. The PINNs, with their inherent capability to capture complex flow patterns, are employed to estimate the aeroacoustic source map. The predicted map obtained from the PINNs is then rigorously compared to the ground truth source map derived from the high-fidelity data. Through these experiments, we demonstrate the remarkable ability of PINNs to effectively estimate the aeroacoustic source map in the presence of noisy and missing data. This validation establishes the potential of PINNs as a powerful tool for aeroacoustic analysis and source characterization. The successful application of PINNs in this study opens up new ways for further advancements in aeroacoustics. By leveraging PINNs we can enhance noise reduction techniques, optimize design processes, and improve the overall efficiency of aerodynamic systems. In conclusion, this research showcases the potential of Physics Informed Neural Networks for accurate aeroacoustic source estimation in scenarios where data quality is compromised. These findings contribute to the growing body of knowledge in aeroacoustics and offer a pathway toward more robust and efficient analysis techniques in the field.