Evidential Uncertainty Quantification in Physics-Informed Neural Networks (PINNs)

This repository contains the implementation and experimental results for our extended abstract on evidential uncertainty quantification in Physics-Informed Neural Networks (PINNs). We introduce a approach that combines Deep Evidential Regression (DER) with PINNs to provide reliable uncertainty estimates for physics-based predictions.

📋 Abstract

We integrate a state-of-the-art method to quantify aleatoric and epistemic uncertainties in physics-informed neural networks and observe that they can be captured effectively while maintaining predictive accuracy.

Setup

1. Clone the repository:

   git clone <repository-url>
   cd pinn-der-ai4x

2. Install dependencies using Poetry:

   poetry install

🚀 Usage

Quick Start

The repository contains two main experimental setups:

1. Burgers' Equation Experiment

cd examples/burgers-kedro-experiment
kedro run

2. Laplace Equation Experiment

cd examples/laplace-kedro-experiment
kedro run

Manual Implementation

For custom PDEs, you can use the core library directly:

from pinn_der_ai4x import PINNTrainer, DERMLP, NIG_REG, NIG_NLL

class CustomPINNTrainer(PINNTrainer):
    def define_neural_network(self):
        return DERMLP(
            insize=input_size,
            outsize=output_size,
            hsizes=hidden_sizes,
            nonlin=torch.nn.Tanh,
        )

    def define_objective_function(self, decision_vars, residual_pde):
        # Define your evidential loss functions here
        nig_nll = NIG_NLL(metric, v, alpha, beta, "nig_nll", scaling)
        nig_reg = NIG_REG(metric, v, alpha, beta, "nig_reg", scaling)
        return [nig_nll, nig_reg]

📊 Experimental Results

Burgers' Equation

• Domain: x ∈ [-1, 1], t ∈ [0, 1]
• Viscosity: ν = 0.01/π
• Network: 2 hidden layers with 32 neurons each
• Activation: Tanh

Laplace Equation

• Domain: 2D rectangular domain
• Boundary Conditions: Mixed Dirichlet/Neumann
• Network: 2 hidden layers with 30 neurons each
• Activation: SiLU

🔬 Methodology

Deep Evidential Regression (DER)

Our implementation uses the Normal Inverse Gamma (NIG) distribution to model:

• μ: Mean prediction
• ν: Evidence parameter (aleatoric uncertainty)
• α: Shape parameter (epistemic uncertainty)
• β: Scale parameter

Loss Functions

1. NIG_NLL: Negative log-likelihood loss for evidential learning
2. NIG_REG: Regularization term to prevent overconfidence
3. PDE Residual: Physics-informed constraint

📁 Repository Structure

pinn-der-ai4x/
├── pinn_der_ai4x/           # Core library
│   ├── der_pinn_lib.py      # DER-PINN implementation
│   └── __init__.py
├── examples/
│   ├── burgers-experiment/  # Simple Burgers' equation
│   ├── burgers-kedro-experiment/  # Kedro pipeline for Burgers'
│   ├── laplace-experiment/  # Simple Laplace equation
│   └── laplace-kedro-experiment/  # Kedro pipeline for Laplace
├── pyproject.toml           # Project configuration
└── README.md                # This file

📚 Dependencies

• PyTorch: Deep learning framework
• Neuromancer: Optimization and constraint handling
• NumPy: Numerical computations
• Kedro: Data pipeline management (for experiments)

📄 Citation

If you use this code in your research, please cite our extended abstract:

@inproceedings{
   kai2025quantifying,
   title={Quantifying Uncertainty in Physics-Informed Neural Networks},
   author={Yip Jun Kai and Eduardo de Conto and Arvind Easwaran},
   booktitle={AI4X 2025 International Conference},
   year={2025},
   url={https://openreview.net/forum?id=tXJ2G0g9HM}
}

👨‍💻 Authors & Maintainers

Name	Connect
Yip Jun Kai	LinkedIn, GitHub
Eduardo de Conto	LinkedIn, GitHub
Arvind Easwaran

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Note: This repository accompanies our extended abstract submission to AI4X 2025. For detailed theoretical foundations and experimental results, please refer to the full extended abstract.