1D Heat Diffusion PINN Model

Traditional numerical methods struggle with complex boundary conditions, noisy or incomplete data, and high computational cost. Physics informed neural networks (PINNs) were designed to overcome these issues by embedding the governing equations directly into the loss function of a neural network, allowing the model to “learn” both from data and physics simultaneously.

Overview

Using a Physics-Informed Neural Network (PINN), we tackle an inverse problem involving the one dimensional heat equation, a fundamental partial differential equation governing thermal diffusion.   

In this model, we attempt to discover the underlying thermal behavior directly from data, while ensuring that the learned solution obeys the physics of heat transfer everywhere in the domain. The main contributions of this work are:

• Formulate a PINN for the one-dimensional heat equation with unknown constant diffusivity, using sparse and noisy synthetic temperature data.

• Demonstrate that the PINN successfully recovers both the full temperature field and the underlying diffusivity parameter alpha with high accuracy.

• Design a baseline neural network trained only on data, without any physics constraints, and show that the PINN outperforms this baseline by approximately 99% in mean-squared error.

• Perform post-processing diagnostics based on local effective diffusivity and PDE residuals, to assess physical consistency, and compare the PINN to the baseline model.

Training Objective

Our goal is to simultaneously learn:

The neural network parameters that represent the temperature field, and the physical parameter (thermal diffusivity), treated as a trainable variable. 

Such that the network fits the observed temperature data:

• The heat equation residual is minimized throughout the domain 

• The initial and boundary conditions are satisfied 

• We correctly discover the underlying thermal diffusivity governing the system

The PINN enforces the governing PDE, initial condition, boundary conditions, and data fidelity through a weighted loss function composed of four terms.

1. PDE Loss: The PDE residual

2. Initial Condition Loss: Given the initial condition 

3. Boundary Condition Loss: Assuming Dirichlet boundary conditions 

4. Data Loss: Given noisy temperature measurements 

Total Loss: The complete PINN loss function is a weighted sum of all above components.

Results

In our experiments,

• The PINN accurately reconstructed the temperature field from sparse, noisy data.

• The learned  thermal diffusivity parameter: alpha_learned = 0.197777 

•  Local estimates of alpha_hat formed a tight distribution around the truth.

• PDE residuals were centered near zero across the entire domain.

Compared to a baseline neural network trained only on data:

• • the PINN reduced prediction error by ≈ 99%

  • the baseline produced unstable, noisy estimates of  alpha_hat

  • the PINN produced a smooth, physically consistent solution