Physics-Informed Neural Networks with Passivity Constraints

Enhancing inverse parameter identification in nonlinear dynamical systems using Physics-Informed Neural Networks (PINNs), thermodynamic passivity constraints, and bootstrap ensemble uncertainty quantification.

This repository presents a complete experimental pipeline for investigating stability, identifiability, and calibration behavior in PINN-based inverse modeling of the nonlinear pendulum.

Highlights

• Inverse parameter recovery under sparse velocity observations (100 measurements)
• Passivity constraints enforcing physically consistent energy dissipation
• 25-model bootstrap ensembles for uncertainty quantification
• 20,000 training epochs per model
• Robustness grids and calibration analysis
• Fully reproducible experimental pipeline

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Research Questions Investigated

• Does passivity regularization stabilize inverse parameter recovery?
• Can bootstrap ensembles provide calibrated uncertainty estimates in PINNs?
• How severe is damping parameter non-identifiability under sparse observations?

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📊 Key Results

Parameter Estimation (Noisy Case, σ=0.01)

Method	g error	L error	c error	Trajectory RMSE	Energy Drift
Standard PINN	0.04%	22.9%	1032%	0.327	0.00199
Passivity PINN	2.12%	9.06%	696%	0.327	0.00058
Ensemble (25)	1.44%	13.3%	768%	0.329	0.00061

Key Findings

• Passivity improves conservative parameters (g, L) by stabilizing estimates
• Damping estimation is catastrophically ill-posed (700–2100% errors) — a fundamental identifiability challenge under sparse observations
• Ensemble UQ is severely miscalibrated (8.7% coverage vs. 95% target)
• Bias dominates over variance — systematic errors are the primary failure mode

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🗂️ Repository Structure

pinn_passivity_paper/
├── src/                          # Main source code
│   ├── data/
│   │   ├── generator.py          # Data generation (analytical + nonlinear solvers)
│   │   └── utils.py              # Time grids, noise, batching
│   ├── baseline/
│   │   ├── linear_small_angle.py # Analytical solutions
│   │   ├── nonlinear_rk.py       # RK4 and solve_ivp
│   │   └── plots_baseline.py     # Baseline plotting
│   ├── models/
│   │   ├── pinn_inverse.py       # PINN architecture with Fourier features
│   │   ├── losses.py             # Physics, IC, passivity losses
│   │   ├── train_inverse.py      # Training loop with TensorBoard
│   │   ├── dissipation_net.py    # NN for nonparametric damping
│   │   └── ensemble.py           # Bootstrap ensemble implementation
│   ├── analysis/
│   │   ├── metrics.py            # RMSE, energy drift, coverage, ECE
│   │   └── tables_figs.py        # Figure/table generators
│   ├── experiments/
│   │   ├── exp_baseline.py       # Baseline experiments
│   │   ├── exp_inverse_single.py # Single PINN runs
│   │   ├── exp_inverse_ens.py    # Ensemble experiments
│   │   └── grids.py              # Robustness study grids
│   ├── configs/
│   │   ├── default.yaml          # All hyperparameters
│   │   └── config_loader.py      # Config management
│   └── viz/
│       └── style.py              # Publication-quality plotting
├── scripts/
│   ├── run_all.sh                # Full pipeline orchestration
│   └── generate_final_study.py  # Automated analysis report
├── tests/
│   ├── test_data.py
│   ├── test_losses.py
│   ├── test_models.py
│   └── test_metrics.py
├── outputs/                      # Generated results
│   ├── baseline/
│   ├── inverse_single/
│   ├── ensemble/
│   ├── summaries/
│   ├── FINAL_STUDY.md
│   └── *.csv, *.png, *.json
├── report/
│   ├── COMPLETE_REPORT.tex       # Full LaTeX report (~40 pages)
│   ├── figures/
│   └── *.md
├── requirements.txt
├── pyproject.toml
├── Makefile
└── README.md

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🚀 Quick Start

1. Install Dependencies

python3 -m venv venv
source venv/bin/activate  # Windows: venv\Scripts\activate
pip install -r requirements.txt

Required packages: torch, numpy, scipy, matplotlib, pandas, tqdm, tensorboard, pyyaml, pytest, ruff, black

2. Run the Full Pipeline

bash scripts/run_all.sh --full

# Or step-by-step:
make setup
make test
make run-baseline
make run-inverse
make run-ensemble

3. View Results

tensorboard --logdir outputs/
cat outputs/FINAL_STUDY.md
open outputs/analysis/*.png

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🧪 Experiments

Baseline

python -m src.experiments.exp_baseline

Single Inverse PINN

python -m src.experiments.exp_inverse_single \
    --n-epochs 20000 \
    --n-sparse 100 \
    --noise 0.01 \
    --use-velocity-obs true

Ensemble UQ

python -m src.experiments.exp_inverse_ens \
    --n-models 25 \
    --n-epochs 20000 \
    --use-passivity true

Robustness Grid

python -m src.experiments.run_grid --full

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🔬 Configuration

Edit configs/default.yaml:

physics:
  g: 9.81
  L: 1.0
  c: 0.05

time:
  t_start: 0.0
  t_end: 10.0
  n_points_dense: 1000
  n_points_sparse: 100

model:
  hidden_dims: [32, 32, 32]
  activation: tanh
  n_fourier_features: 6

training:
  n_epochs: 20000
  lr: 0.001
  optimizer: adam

loss_weights:
  data: 1.0
  velocity: 1.0
  physics: 10.0
  ic: 1.0
  passivity: 1.0

ensemble:
  n_models: 25
  bootstrap: true

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🎯 Reproducibility

Seed: 1337 (fixed throughout). All experiments use fixed random seeds, deterministic algorithms, logged hyperparameters, and saved checkpoints.

export PYTHONHASHSEED=1337
bash scripts/run_all.sh --full

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🧪 Testing

pytest                         # All tests
pytest tests/test_losses.py -v # Specific module
pytest --cov=src tests/        # With coverage

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📝 Report

cd report/
pdflatex COMPLETE_REPORT.tex
pdflatex COMPLETE_REPORT.tex

The report contains 40+ pages of analysis, 20+ figures, 15+ result tables, and full mathematical derivations. Compatible with Overleaf.

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📚 References

1. Raissi et al. (2019) — Physics-Informed Neural Networks, Journal of Computational Physics
2. Karniadakis et al. (2021) — Physics-Informed Machine Learning, Nature Reviews Physics
3. Wang et al. (2021) — Gradient Pathologies in PINNs, SIAM Journal on Scientific Computing
4. Yang et al. (2021) — Bayesian PINNs, Journal of Computational Physics

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📄 Academic Context

This work was completed as part of MA-515 at IIT Ropar.

Authors: Pranav Singh · Prashant Singh · Nishit Soni · Jaskaran Singh · Ishwar Sanjay · Harshdeep

@techreport{singh2024pinn,
  title={Enhancing PINN-Based Inverse Modeling of the Nonlinear Pendulum
         Using Passivity Constraints and Ensemble UQ},
  author={Singh, Pranav and Singh, Prashant and Soni, Nishit and
          Singh, Jaskaran and Sanjay, Ishwar and Harshdeep},
  institution={Indian Institute of Technology Ropar},
  year={2024},
  type={Course Project Report},
  number={MA-515}
}

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Last Updated: November 2025