A simple PyTorch implementation of a Physics-Informed Neural Network (PINN) to solve a first-order pharmacokinetic ODE. This project demonstrates how to enforce physical laws (differential equations) directly into the loss function of a neural network.

We model the concentration of a drug in the bloodstream, $C(t)$, which decays over time following a simple first-order differential equation:

$$ \frac{dC}{dt} = -kC $$

Subject to the initial condition:

$$ C(0) = C_0 $$

Where:

• $C(t)$ is the drug concentration at time $t$.
• $k$ is the elimination rate constant.
• $C_0$ is the initial concentration.

Instead of training purely on data points, the neural network $NN(t; \theta)$ minimizes a composite loss function:

1. Boundary Loss: Ensures $NN(0) \approx C_0$.
2. Physics Loss: Enforces the residual $\left( \frac{dNN}{dt} + k \cdot NN \right)^2 \approx 0$ across the time domain using automatic differentiation.

Run the standalone script to train the model and generate the loss plot and concentration animation:

python drug_pinn.py

This will produce:

• drug_pinn_concentration.gif: An animation of the learned solution.
• drug_pinn_loss.png: The training loss history.

For more reading, see:

Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707.

Created by Zara Darcy and assisted by Cursor AI.