Partial Differential Equations (PDEs) is still lacking. This study introduces PIN-Nacle, a benchmarking tool designed to fill this gap.

This paper studies quasi-Newton methods for solving nonlinear equations. We propose block variants of both good and bad Broyden's methods, which enjoy explicit local superlinear convergence rates. Our block good Broyden's method has a faster condition-number-free convergence rate than existing Broyden's methods because it takes the advantage of multiple rank modification on Jacobian estimator. On the other hand, our block bad Broyden's method directly estimates the inverse of the Jacobian provably, which reduces the computational cost of the iteration. Our theoretical results provide some new insights on why good Broyden's method outperforms bad Broyden's method in most of the cases. The empirical results also demonstrate the superiority of our methods and validate our theoretical analysis.

In the reminder of the paper we will try whenever possible to formally introduce functions by clearly stating domain and co-domain.

Dynamics forecasting refers to the task of predicting the future behavior of a dynamic system, involving learning the underlying dynamics governing the system's evolution to make accurate predictions

Interacting systems are prevalent in nature.