[Paper Review] Physics-Informed Extreme Theory of Functional Connections Applied to Data-Driven Parameters Discovery of Epidemiological Compartmental Models

Enrico Schiassi, Andrea D’Ambrosio+3 more|arXiv (Cornell University)|Aug 12, 2020

Model Reduction and Neural Networks18 references9 citations

TL;DR

This paper proposes a physics-informed extreme theory of functional connections (X-TFC) framework to solve inverse problems in epidemiological compartmental models (SIR, SEIR, SEIRS) by discovering unknown parameters from noisy and unperturbed data. By embedding the governing ODEs as hard constraints via functional interpolation, X-TFC enables fast, accurate, and robust parameter estimation with low computational cost, outperforming standard regression by avoiding overfitting even under significant noise levels.

ABSTRACT

In this work we apply a novel, accurate, fast, and robust physics-informed neural network framework for data-driven parameters discovery of problems modeled via parametric ordinary differential equations (ODEs) called the Extreme Theory of Functional Connections (X-TFC). The proposed method merges two recently developed frameworks for solving problems involving parametric DEs, 1) the Theory of Functional Connections (TFC) and 2) the Physics-Informed Neural Networks (PINN). In particular, this work focuses on the capability of X-TFC in solving inverse problems to estimate the parameters governing the epidemiological compartmental models via a deterministic approach. The epidemiological compartmental models treated in this work are Susceptible-Infectious-Recovered (SIR), Susceptible-Exposed-Infectious-Recovered (SEIR), and Susceptible-Exposed-Infectious-Recovered-Susceptible (SEIR). The results show the low computational times, the high accuracy and effectiveness of the X-TFC method in performing data-driven parameters discovery of systems modeled via parametric ODEs using unperturbed and perturbed data.

Motivation & Objective

• Address the challenge of ill-posed inverse problems in epidemiological modeling, where parameter estimation is sensitive to data noise and model uncertainty.
• Develop a deterministic, physics-informed framework that integrates the Theory of Functional Connections (TFC) with physics-informed neural networks (PINNs) to improve robustness and accuracy in parameter discovery.
• Demonstrate the effectiveness of the X-TFC framework in estimating parameters for SIR, SEIR, and SEIRS models using both clean and noisy data, ensuring stability and convergence.
• Reduce computational time while maintaining high accuracy in parameter estimation, enabling real-time updates in emergency response scenarios.
• Validate the method’s ability to avoid overfitting to noisy data by comparing against standard regression baselines.

Proposed method

• Integrate the Theory of Functional Connections (TFC) with extreme learning machine (ELM) optimization to construct parametrized solutions that satisfy initial conditions and ODE constraints exactly.
• Embed the governing parametric ordinary differential equations (ODEs) of compartmental models (SIR, SEIR, SEIRS) as hard constraints in the neural network loss function, ensuring physical consistency.
• Use a deterministic optimization framework to minimize the residual of the ODE system and the data misfit, treating parameters as learnable variables.
• Apply the extreme learning machine (ELM) algorithm to accelerate training by fixing input weights and biases, enabling fast convergence with minimal hyperparameter tuning.
• Construct constrained functional forms that inherently satisfy initial conditions and ODE dynamics, reducing the search space and improving training stability.
• Train the network using both unperturbed and perturbed datasets with increasing noise levels (δ = 0.1, 1, 3) to evaluate robustness and generalization.

Experimental results

Research questions

• RQ1 Can the X-TFC framework accurately estimate unknown parameters in SIR, SEIR, and SEIRS models from noisy observational data?
• RQ2 How does the X-TFC method compare to standard regression in avoiding overfitting when data is corrupted by noise?
• RQ3 What is the computational efficiency of X-TFC in solving inverse problems for epidemiological models with varying noise levels?
• RQ4 To what extent does embedding the physics of the ODE system into the network architecture improve solution stability and accuracy?
• RQ5 Can X-TFC maintain high accuracy in parameter estimation even when the number of data points is limited or the data is significantly perturbed?

Key findings

• For the SIR model with no noise, X-TFC achieved exact parameter recovery in 3 iterations with a CPU time of 0.004 seconds, and absolute errors of 0 for all parameters.
• With 10% noise (δ = 0.1), X-TFC estimated β = 0.2971 (error: 2.9×10⁻³), γ = 0.5996 (error: 3.7×10⁻⁴), σ = 0.9005 (error: 4.9×10⁻⁴), and R₀ = 0.4955 (error: 4.5×10⁻³), demonstrating high accuracy.
• For the SEIR model with δ = 3, X-TFC achieved absolute errors below 5×10⁻² for all parameters, with β = 0.2130 (error: 8.7×10⁻²), γ = 0.5878 (error: 1.2×10⁻²), σ = 0.9141 (error: 1.4×10⁻²), and R₀ = 0.3624 (error: 1.4×10⁻¹), showing robustness to high noise.
• For the SEIRS model with δ = 3, the method achieved absolute errors below 6×10⁻² for all parameters, including ζ = 0.5576 (error: 5.8×10⁻²), indicating reliable performance under high perturbation.
• The X-TFC framework required only 110 iterations and 1.36 seconds to estimate parameters under δ = 3 in the SEIRS model, demonstrating scalability and low computational cost.
• X-TFC consistently outperformed standard regression by avoiding overfitting to noisy data, as evidenced by the small parameter errors despite high noise levels.