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Enhancing Parametric PINN Training: A Synergistic Approach with Weighted Curricula and Learning Rate Scheduling

Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm for solving differential equations by embedding physical laws directly into the loss function of a neural network. A particularly compelling application is in solving parametric inverse problems, where the goal is to infer physical parameters (e.g., viscosity, conductivity) from observational data. A common and effective methodology involves a two-stage process: first, training a general parametric model over a range of the parameter, and second, using this pre-trained model as a prior to rapidly identify the specific parameter value that best explains a new set of observations.

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Overcoming Generalization Challenges in 2D Burgers' Equation with Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm in "scientific machine learning," enabling the integration of physical laws, expressed as Partial Differential Equations (PDEs), into neural network training (Wang et al., 2021, p. 1; Lu et al., 2021, p. 208). While PINNs offer a promising avenue for solving complex, non-linear systems like the 2D Burgers' equation, ensuring robust generalization remains a critical challenge, particularly for solutions exhibiting multi-scale features, anisotropic behavior, or in data-poor environments. The 2D Burgers' equation is a fundamental non-linear PDE often used as a benchmark for such challenges (Raissi et al., 2019, p. 686). This post explores several cutting-edge approaches aimed at enhancing PINN generalization capabilities, directly addressing the underlying limitations that hinder their performance.

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Overcoming Challenges in Physics-Informed Neural Networks (PINNs): Gradient Optimization for Inverse Problems

Physics-Informed Neural Networks (PINNs) represent a promising approach for solving complex Partial Differential Equations (PDEs) and inverse problems, such as determining hidden parameters of a physical system – for instance, viscosity (nu) in a 2D Burgers equation. However, the application of PINNs presents inherent challenges. A recent study by Wang et al. (2020) delves into these limitations and proposes innovative solutions that can significantly enhance PINN performance.