tuberculosis-pinn-solver-numerical-solver
Physics-Informed Neural Network for Tuberculosis Modeling
Welcome! This project explores the use of Physics-Informed Neural Networks to solve a tuberculosis ODE model.
This project presents a computational solution to a tuberculosis (TB) compartmental model using:
- Classical numerical solvers
- Physics-Informed Neural Networks (PINNs)
The PINN approach integrates the system of ODEs into the neural network training process, allowing for continuous and accurate approximations of disease dynamics over time.
Project Overview
We modeled the spread and progression of TB using a system of four ODEs representing the following compartments:
- S(t): Susceptible individuals
- E(t): Exposed (infected but not yet infectious)
- I(t): Infectious individuals
- L(t): Latent or partially treated individuals
Two solution methods were implemented:
- Numerical integration methods (e.g., Runge-Kutta-Fehlberg, Heun’s Method)
- Physics-Informed Neural Network (PINN) trained using PyTorch
Methods Used
Numerical Solvers:
- Heun’s Method (Predictor-Corrector)
- Classical Runge-Kutta 4th Order (RK4)
- Adaptive Runge-Kutta Method
Machine Learning Solver:
- Physics-Informed Neural Networks (PINNs)
- Implemented using PyTorch
- Trained on ODE physics, initial conditions, and data points
- Supports extrapolation from t=0 to t=100
Results
| Method | Most Accurate | Fastest Execution | Extrapolation Support |
|---|---|---|---|
| Heun | ✗ | Fast | ✗ |
| RK4 | ✓ Very Accurate | Moderate | ✗ |
| Adaptive RK | ✓ Accurate | Fastest | ✗ |
| PINN | ✓ Accurate | Fast (post-training) | ✓YES |
Mean Squared Error (MSE) per Method:
| Compartment | Heun | RK4 | Adaptive RK | PINN |
|---|---|---|---|---|
| S | 3.6878 | 0.0001 | 0.0163 | 1.37 |
| E | 3.3949 | 0.0001 | 0.0151 | 1.27 |
| I | 0.0227 | 0.0000 | 0.0001 | 0.009 |
| L | 0.000002 | 0.0000 | 0.0000 | 0.000001 |
- PINN outperformed numerical integration in speed (7900% faster).
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Files
📁 adaptive_quad/ → Contains adaptive_quad.py – Adaptive Quadrature integration method
📁 figures/ → Plots and result visualizations (MSE curves, method comparisons, etc.)
📁 gauss-methods/ → Contains gauss_methods.py – Gauss quadrature-based solver
📁 heun_non_self_start/ → Contains heun_non_self_start.py – Heun’s method for solving ODEs
📁 romberg_integration/ → Contains romberg_integration.py – Romberg integration-based solver
📁 runge-kutta-method/ → Contains runge_kutta_method.py – Classic 4th order Runge-Kutta method
Numerical_Project_Full.ipynb → Jupyter notebook version that includes all code and result analysis
Each Python script contains a solver implementation for one of the numerical methods used in this study, with consistent function structures and outputs and plots for modular testing and comparison
Trials
1. Attempted Method: Romberg Integration
What is Romberg Integration?
Romberg Integration is a numerical method used to estimate definite integrals with high precision. It combines the Trapezoidal Rule with Richardson Extrapolation to refine the accuracy of the result over successive approximations.
It starts from the basic trapezoidal estimate:
Tₙ = (h / 2) × [f(a) + 2 × ∑ f(xᵢ) + f(b)]
Then applies Richardson extrapolation recursively to eliminate lower-order error terms:
R(n, m) = [4ᵐ × R(n, m−1) − R(n−1, m−1)] / (4ᵐ − 1)
This builds a Romberg table where the bottom-right value is the most accurate approximation of the integral.
Why Romberg Integration Did Not Work in Our Case
Although Romberg is powerful for integrating functions of a single variable (usually time), it is not suitable for our TB ODE model because:
- Our ODE system has the form:
dy/dt = f(y)
where f(y) does not depend on time t explicitly.
- Romberg Integration is most effective when integrating:
∫ f(t) dt
with function values changing significantly over t.
- In our model, since the right-hand side
f(y)is independent oft, the integration essentially becomes:
yₙ₊₁ ≈ yₙ + f(yₙ) × dt
which reduces to Euler’s method, but with extra computational cost and no added accuracy.
2.Gauss Quadrature: Why It Was Considered and Why It Was Excluded?
What is Gauss Quadrature?
Gauss Quadrature, specifically the Gauss–Legendre method, is a high-accuracy numerical integration technique. It works by evaluating a function at carefully selected points (called Gauss nodes) within an interval and summing the weighted values to approximate a definite integral. This method is often preferred over basic rules like trapezoidal or Simpson’s because it achieves much higher accuracy with fewer evaluations when applied to smooth functions.
In the context of solving ordinary differential equations (ODEs), Gauss Quadrature can be used in implicit integration schemes, particularly in stiff systems or when high-order precision is required.
Why We Considered Using It
During the early stages of the project, we explored whether applying Gauss Quadrature to the TB model’s ODE system could yield advantages in speed or accuracy. The appeal was based on Gauss Quadrature’s well-known ability to:
- Accurately approximate definite integrals
- Converge rapidly with fewer evaluations
- Reduce local truncation error in some numerical schemes
Why We Ultimately Did Not Use It
In such systems, the derivatives depend only on the state variables (S, E, I, L), not on time (t) directly. Gauss Quadrature methods, however, are designed for integrating functions with respect to an independent variable, typically time. Their advantage comes from evaluating the function at strategically chosen points across the time interval to approximate:
But because f in our case is not a function of t, Gauss Quadrature does not deliver any real performance or accuracy benefits. It ends up behaving similarly to basic methods like Euler’s, relying primarily on the step size dt for accuracy. Additionally, the method introduces extra computational overhead without a meaningful return.
Conclusion
- The system is autonomous (no explicit time dependence),
- The integrals in question do not benefit from Gauss’s strategic sampling,
- The method adds complexity without enhancing solution quality.
For these reasons, we decided to exclude Gauss Quadrature from our final set of evaluated methods and focus instead on Heun’s method, RK4, Adaptive RK, and Physics-Informed Neural Networks (PINNs), which are better suited for this type of problem.
👥 Team Members
- [rowida moahemd ]
- [zyad hamed ]
- [Ali ahmed gad ]
- [yousef samy ]
- [yousef mortada ]
- [Mohamed ward ]
- [malak sherif ]
- [yomna sabry ]
- [mohamed nasser ]
- [ Abdelrahman Emad ]
- [farah yehia ]
- [yousef magdy]
References
[1] Schiesser, W. E. (2014). Differential equation analysis in biomedical science and engineering.
[2] Applied Numerical Methods with Python for Engineers and Scientists(1st ed.). (2021). McGraw-Hill Higher Education.
[3] S. Kanwal, M. K. Siddiqui, E. Bonyah, K. Sarwar, T. S. Shaikh, andN. Ahmed, “Analysis of the epidemic biological model of tuberculosis(TB) via numerical schemes,” Complexity, vol. 2022, Art. ID 5147951,13 pp., Mar. 2022, doi: 10.1155/2022/5147951.
[4] E. D. Tadesse, M. A. Bhat, and A. Ayele, “A deterministic compartment model for analyzing tuberculosis dynamics considering vaccination and reinfection,” Heliyon, vol. 9, no. 10, p. e19674, 2023, doi:10.1016/j.heliyon.2023.e19674
[5] S. Side, A. M. Utami, S. Sukarna, and M. I. Pratama, “Numerical solution of SIR model for transmission of tuberculosis by Runge–Kutta method, Journal of Physics: Conference Series, vol. 1040, p. 012021,2018
[6] Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2017). Physics Informed Deep Learning (Part I): Data-driven solutions of nonlinear partial differential equations. arXiv (Cornell University).
[7] Rasheed, A. (2024). Modeling Dynamical Systems with Physics Informed Neural Networks with Applications to PDEConstrained Optimal Control Problems.https://ntnuopen.ntnu.no/ntnuxmlui/handle/11250/3130805?show=full
[8] Pal, B., Rahaman, R., et.al. (2025). A deep learning approach to model and predict tuberculosis using PINNs.
[9] Lu, L., Meng, X., Mao, Z., & Karniadakis, G. E. (2019). DeepXDE: A deep learning library for solving differential equations. arXiv.
[10] Walters, S. J., Turner, R. J., & Forbes, L. K. (2022). A COMPARISON OF EXPLICIT RUNGE–KUTTA METHODS. The ANZIAM Journal,64(3), 227–249.
- Cairo University Numerical Methods materials (Spring 2025)
Contact
rowida.mohamed04@eng-st.cu.edu.eg