This talk presents Transformed Generative Pre-Trained Physics-Informed Neural Networks (TGPT-PINN), a novel architecture for nonlinear model order reduction (MOR) of transport-dominated partial differential equations (PDEs) within an MOR-integrated PINNs framework. Building upon the GPT-PINN's network-of-networks design for snapshot-based model reduction, TGPT-PINN introduces a paradigm-shifting approach to handle parameter-dependent discontinuities through two key innovations: (1) a shock-capturing loss component and (2) a parameter-adaptive transform layer, collectively overcoming linear MOR's limitations in transport-dominated regimes. The method's efficacy is demonstrated through multiple non-trivial parametric PDE test cases. Furthermore, we present the Entropy-enhanced TGPT-PINN (EGPT-PINN), which achieves remarkable accuracy in solving Burgers' and Euler equations' viscosity solutions using minimal neurons—without relying on a priori knowledge of governing equations or initial conditions. By augmenting the loss function with a model-data mismatch term, EGPT-PINN demonstrates superior robustness in inverse problem solving compared to both vanilla PINNs and existing entropy-enhanced variants. The talk will highlight these architectures' theoretical foundations, implementation strategies, and benchmark results across forward and inverse problems.

With the commencement of the LIGO-Virgo-KAGA Collaboration's fourth observing run, the field of gravitational wave (GW) physics is uniquely poised to collect even more precise data from compact binary coalescences. Consequently, we will soon be able to perform even more stringent tests of general relativity. One such test that is particularly promising is trying to verify black hole perturbation theory's predictions using the ringdown excitations imprinted on the merger's remnant black hole. Doing so, however, requires that we have a robust understanding of these excitations through numerical relativity simulations as well as a means to model them across binary black hole parameter space. In this talk, I will highlight recent advancements regarding our understanding of quasi-normal modes (QNMs) that have made performing this test even more alluring. In particular, I will show how QNMs probe the intrinsic geometry of black holes and even reveal information about the nonlinear nature of Einstein's equations. Following this, I will then explain how we can model QNMs using Gaussian process regression and will provide updates on the current status of these models and how they can be used in gravitational wave data analysis to probe both astrophysics as well as the nature of gravity.

Black hole binaries with small mass ratios will be critical targets for the forthcoming Laser Interferometer Space Antenna (LISA) mission. They also serve as useful tools for studying the dynamics of compact binary systems and the gravitational waves they emit. Using an eccentric Ori-Thorne procedure, we build trajectories that describe the full inspiral and plunge of a small body on an initially eccentric orbit of a Kerr black hole. We then calculate the gravitational waves associated with these trajectories with a code that solves the Teukolsky equation in the time domain. The final cycles of these waveforms are a superposition of Kerr quasinormal modes. In this work, we examine the impact of orbital eccentricity and its associated phase parameters on quasinormal mode excitation. We find that a binary’s eccentricity during its last orbital revolutions has notable signatures on the relative mode excitation. If these signatures from the small-mass-ratio regime are present at less extreme mass ratios, then measuring gravitational ringdown modes from binary black hole mergers may provide key insight into a binary’s geometry. A diagnostic of eccentricity in the late waveform could be a valuable tool for understanding the systems that create these signals for next-generation detectors like LISA, as well as the ground-based detectors operating today.

This talk highlights recent advancements in Physics-Informed Machine Learning (PIML), focusing on innovations in Kolmogorov-Arnold Networks (KANs). Since their introduction in 2017, Physics-Informed Neural Networks (PINNs) have become essential tools for solving differential equations from sparse data. Advances in network architectures, adaptive refinement, domain decomposition, and weighting strategies have further enhanced their performance. Among these, Physics-Informed Kolmogorov-Arnold Networks (PIKANS), based on Kolmogorov’s 1957 representation model, offer a robust alternative to traditional PINNs.

A key application of these developments is the Artificial Intelligence Velocimetry-Thermometry (AIVT) method, which infers continuous, differentiable temperature fields in turbulent convection from sparse 3D velocity data. AIVT achieves high accuracy, validated against a unique experimental dataset combining Particle Image Thermometry and Lagrangian Particle Tracking. The method demonstrates fidelity consistent with theoretical predictions and performance comparable to direct numerical simulations, offering novel insights into turbulent flows.

Additionally, the talk introduces the Kurkova-Kolmogorov-Arnold Network (KKAN), a two-block architecture inspired by the Kolmogorov-Arnold theorem and Kurkova’s principles. KKAN integrates multi-layer perceptrons with flexible basis functions, achieving superior performance in function approximation, physics-informed learning, and operator tasks. Insights into KKAN’s learning dynamics, analyzed through information bottleneck theory, reveal trends across representation models and tasks, highlighting correlations between geometric complexity and generalization capabilities. These findings provide valuable perspectives on the behavior and utility of advanced PIML architectures.