Student Seminar: Physics-Informed Neural Networks (PINNs): Application to Liquid State Theory

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for solving differential and integral equations in regimes where data are sparse but physical laws are well established. In liquid-state theory, the structural properties of homogeneous fluids are governed by the Ornstein–Zernike (OZ) equation, which couples total and direct correlation functions and traditionally requires approximate closure relations and iterative Fourier-transform techniques for its solution. These numerical approaches often suffer from slow convergence, instability, and failure near critical states.​

In this talk, I will discuss how PINN provides an alternative approach for modeling liquid correlation functions by directly embedding closure relations and the OZ equation into a loss function. This allows the network to learn correlation functions such as h(r), c(r), and γ(r) from the physics itself. The PINN architecture integrates real-space and reciprocal-space representations using forward and inverse Fourier transforms. The results demonstrate superior stability close to the critical state and excellent agreement with iterative Fourier-transform solutions for PY and HNC closures. PINNs provide accurate, robust, and physically consistent predictions for liquid-structure correlation functions, suggesting a promising alternative to conventional integral-equation solvers in statistical mechanics.​

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