The Ornstein–Zernike (OZ) equation is the fundamental equation for pair correlation function computations in the modern integral equation theory for liquids. In this work, machine learning models, notably physics-informed neural networks and physics-informed neural operator networks, are explored to solve the OZ equation. The physics-informed machine learning models demonstrate great accuracy and high efficiency in solving the forward and inverse OZ problems of various bulk fluids. The results highlight the significant potential of physics-informed machine learning for applications in thermodynamic state theory.

Pseudospectral method has the merit of high accuracy and the defects of simple geometry suitability and low computational efficiency. To remedy the two defects, a multidomain multigrid Chebyshev pseudospectral method is proposed and validated through the numerical solution of two-dimensional incompressible Navier-Stokes equations in the primitive variable formulation. To facilitate the implementation of the multidomain multigrid method, the IP_N-IP_N method is utilized to approximate the velocity and pressure with the same degree of Chebyshev polynomials within each subdomain, and an interface/boundary condensation method is developed to implement the pseudospectral operators of multigrid at the interface/boundary of subdomains. The accuracy and efficiency of the proposed method are first validated by numerical solutions of the lid-driven cavity problem. The numerical results are in good agreement with the benchmark solutions, and the speeding up of multigrid is 4–9 compared against the single grid. Then the capability of the proposed method for even more complex geometries with a close/open boundary is demonstrated by numerical solutions of several typical problems. The proposed method is quite generic and can be extended to the high accuracy and efficiency solution of three-dimensional incompressible/compressible, unsteady/steady fluid flows and heat transfer problems.

Physics-informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and efficiency. In this work, we introduce a hybrid adaptive sampling and weighting method to enhance the performance of physics-informed neural networks (PINNs). The adaptive sampling component identifies training points in regions where the solution exhibits rapid variation, while the adaptive weighting component balances the convergence rate across training points. Numerical experiments show that applying only adaptive sampling or only adaptive weighting is insufficient to consistently achieve accurate predictions, particularly when training points are scarce. Since each method emphasizes different aspects of the solution, their effectiveness is problem dependent. By combining both strategies, the proposed framework consistently improves prediction accuracy and training efficiency, offering a more robust approach for solving PDEs with PINNs.

Accurate 3D fish localization supports understanding of movement, behavior, and habitat use. Traditional solvers such as approximate maximum likelihood (AML) are vulnerable to multipath, time-of-arrival (TOA) measurement errors, and synchronization errors, while Yet Another Positioning Solver (YAPS) can degrade with sparse detections. We introduce a physics-informed neural network (PINN) solver that models trajectories as continuous, physics-constrained functions, minimizes the negative log-likelihood of TOA noise, and quantifies uncertainty via ensemble training. We validated the approach using synthetic experiments and field deployments in the Salish Sea and a dam tailrace. Under synthetic scenarios with abrupt turns, accelerations, and gaps, the solver achieved sub-meter root mean square (RMS) errors of 0.30–0.80 m and >83% efficiency. In the Salish Sea, RMS error was 1.17–2.24 m with 23%–76% efficiency, reducing RMS by ~60% versus AML, and reducing RMS by >75% versus YAPS. In the tailrace, RMS error was 0.81–0.93 m with >92% efficiency, reducing RMS by ~35% versus AML and by >80% versus YAPS. The proposed PINN solver delivers accurate localization across challenging conditions, enabling fine-scale ecological monitoring.