A HYBRID FSIR MODEL WITH PARAMETER EMBEDDING BASED ON MACHINE LEARNING METHODS

Abstract

The paper proposes a hybrid model FSIR (Fractional Susceptible-Infected-Recovered), which combines fractional dynamics with adaptive parameterization through neural networks.

The basic dynamics is described by a fractional FSIR system with a Caputo derivative of the order α=0.9. The simulation time interval t∈[0,100] is divided into 500 uniformly spaced points; the grid pitch h is calculated as the difference of adjacent points. Initial conditions: S(0)=0.99; I(0)=0.01; R(0)=0. Numerical integration of fractional derivatives was performed by the Adams-Bashforth-Moulton method, which is a predictor-corrector method and is implemented in the code for solving FSIR for different arrays of parameters β(t) and γ(t). The paper discusses two parameterization approaches: MLP for parameter embedding and the PINN (Physics-Informed Neural Network) approach.

Network training was carried out on a full grid for all points simultaneously with the Adam optimizer and an initial learning rate of 0.01. Experiments were carried out with a different number of epochs [100, 500, 1000, 2000].

In the PINN approach we used subnets for β(t), γ(t), and [S,I,R]. with three layers of 32 Tanh-activated neurons. In this approach the loss function combines the MSE with the data responsible for the states and parameters and the physical residual component, which takes into account fractional residues through the L1 approximation of the Caputo derivative. The PINN approach also used the Adam optimizer, learning rate 5e-03 and for some experiments we increased the weight of the physics-based loss term.

The synthetic MLP embedding scenario gave a significant reduction in the error of reproducing the trajectories of epidemic time series according to the SIR model with an increase in the number of epochs. Thus, the average MSE values for the MLP approach decrease from ≈7.04e-04 at 100 epochs to ≈1.4e-05 for 2000 epochs. At 2000 MLP epochs the model reproduces a global SIR trajectory with deviations of less than 1% at almost all points in the time series.

The considered configuration of the PINN model has higher MSE values (~1e-03), e.g. for 2000 epochs 〖MSE〗_total≈1.28e-03. This may indicate the need for additional adjustment of adaptive loss balance, network depth, or the use of combined optimizers to achieve accuracy comparable to the MLP approach

The proposed approach combines the interpretation of the fractional FSIR model with the flexibility of data-driven parameterization. It has been shown that in synthetic tests, MLP embedding achieves low parameter approximation errors (〖MSE〗_total~1e-05), while the PINN approach gives us better physical consistency due to additional optimization, but requires careful tuning for fractional systems and oscillating parameters of the epidemiological process development. We recommend applying a hybrid approach: first, pre-training the MLP model on a wide set of synthetic data with different scenarios, and then fine-tuning using the PINN model on empirical data with the balance of the physics-based loss term in loss function