So basically all these methods to solve ODEs are mostly sequential, that cannot be implemented in parallel.

• Train a Neural Network to predict the output $y(t)$ directly from the input $t$
• Use RK Method once to generate high quality dataset of time-steps and solutions. Then train a neural network to map $t$ → $y(t)$.
• Once trained, the network can predict the state at $t=100$ as fast as it can at $t=0.1$. This is called inference, and it is often orders of magnitude faster than running thousands of sequential RK steps.

Neural Network notes :

PINNs can be used to train here.

PINN - Physics Informed Neural Networks are a type of Neural Networks that integrates know physical laws (like Partial DE) directly into its learning process, allowing it to solve complex scientific engineering problems, approximate solutions to equations, and tackle inverse problems even with limited data by acting as a powerful function approximator.

• By incorporating physics as a regularization term, PINNs ensure predictions respect physical principles

Initial Implementation of both methods in CPP gave these results for a damped oscillation example :

• For a simple cpp simulation, the RK4 method has shown some damping properties than the Euler’s method.
• We can observe the drift in the Euler’s graph compared to the RK4 method.

Moving ahead in Neural Network implementation :

Since RK4 method is stable compared to all the methods, we use RK4 points as ground truth for training the neural network.

Sample Ground Truth values generated using rk4 for a sample damping oscillation example (second order ODE) are attached in results.csv (there are around 200 training points for small scale simulation.)

A basic PINN script that is run on the results.csv as ground truth values is in pinn_solver.py & pinn_solver.ipynb