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+# Physics-Informed Neural Networks (PINNs) for Implicit Runge-Kutta Methods
+
+This project implements Physics-Informed Neural Networks (PINNs) to learn implicit Runge-Kutta (IRK) time-stepping schemes for differential equations, specifically targeting the Lorenz system and advection-diffusion equations.
+
+## Overview
+
+Traditional numerical methods for solving differential equations require explicit computation of intermediate stages in implicit Runge-Kutta schemes, often involving expensive iterative solvers like Newton's method. This project explores using neural networks to directly predict these intermediate stages, potentially accelerating time integration.
+
+### Algorithm Overview
+
+To advance a system state **S** from time **t** to **t + Δt**:
+
+1. **Generate Legendre Points**: Create ~100 Legendre-Gauss-Lobatto quadrature points within the interval (t, t+Δt)
+
+2. **Neural Network Prediction**: Train a neural network to map the initial state **s ∈ S** to all intermediate solutions **s_i ∈ S×N** where **N** is the number of Legendre points. The network essentially approximates the integral of the differential equation over the time interval.
+
+3. **Derivative Computation**: Compute derivatives at each solution point using Newton's method for high accuracy
+
+4. **IRK Integration**: Insert the computed derivatives into the Implicit Runge-Kutta coefficient matrix to obtain the final solution at **t + Δt**
+
+### Key Innovation
+
+The neural network is retrained for each time step, learning to approximate the integral of the differential equation within each specific time interval. Rather than solving the traditional nonlinear system:
+```
+k_i = f(y_n + dt * Σ a_ij * k_j)  for i = 1,...,s
+```
+
+The network directly predicts all IRK stages `{k_1, k_2, ..., k_s}` simultaneously, which are then refined using Newton iteration for guaranteed accuracy.
+
+## Usage
+
+```bash
+# Install dependencies
+pip install -r requirements.txt
+
+# Run main simulation
+python main.py
+
+# Run tests
+pytest .
+```