Boosted physics-informed neural network tracks computer virus spread with vertical transmission

Computer viruses no longer spread only through contact between infected and vulnerable machines. In many realistic scenarios, nodes can be compromised from the moment they join a network, a phenomenon modeled as vertical transmission, in which new devices are initially infected without any exchange with an infectious node. A study published in Discover Artificial Intelligence on 28 August 2026 presents a boosted Physics-Informed Neural Network (PINN) framework designed to solve the nonlinear equations that describe such an e-epidemic, and reports that the approach reproduces reference solutions with low error while offering potential computational savings over classical numerical solvers.

The underlying mathematical model is an SI1I2R system, an epidemic-type framework adapted for computer networks. The total node population is divided into four compartments: susceptible nodes (S), a moderately infectious class (I1), a fully infectious class (I2), and recovered nodes (R). The virus flows from S through I1 to I2 and finally to R, governed by a set of coupled nonlinear ordinary differential equations. A distinctive feature is the vertical transmission parameter sigma, which directly feeds the I1 class without requiring contact, capturing situations where machines are compromised at the outset. The researchers used specific parameter values in their simulations: a node growth rate of 0.03, an attack rate of 0.01, a progression rate from I1 to I2 of 0.095, a recovery rate of 0.90, a vertical transmission rate of 0.03, and death rates reflecting both natural attrition and virus-induced failures.

Classical numerical solvers such as Runge–Kutta methods remain the standard tool for integrating such systems, but the authors note that they can become computationally demanding for large-scale problems and may exhibit limited stability in long-term predictions, because they require discretization of the domain and iterative solution at each step. PINNs offer an alternative: a deep neural network whose loss function embeds the governing differential equations themselves, so that the network is trained not only to fit data but also to satisfy the physics of the system. Introduced by Raissi, Perdikaris, and Karniadakis, the PINN approach uses automatic differentiation to compute the derivatives needed for the physics residuals, and it provides mesh-free approximate solutions with well-behaved analytical gradients, supporting both forward and inverse modeling within a single optimization framework.

The architecture used in this study is a fully connected network with a single input node representing time and four output nodes corresponding to the S, I1, I2, and R populations. It comprises three hidden layers, two with 128 neurons and one with 64, and is optimized with the Adam optimizer over 20,000 epochs using 500 randomly chosen training points within the temporal domain. The composite loss function combines three terms: a residual loss measuring how far the network’s predictions deviate from the governing equations at the training points, an initial-condition loss anchoring the solution to the specified starting values of 100 susceptible, 5 infected in I1, 1 in I2, and 0 recovered nodes, and a data-driven loss comparing predictions with ground-truth values.

The central methodological contribution is what the authors call a boosted training strategy, built around a hybrid Swish–ReLU activation function. The activation is defined as a weighted combination of the smooth, non-monotonic Swish component and the piecewise-linear ReLU component, with a weighting parameter controlling the balance between them. The Swish part preserves differentiability and non-zero gradients across the input domain, which matters when approximating continuous solutions of differential equations, while the ReLU part strengthens activation in the positive region and accelerates convergence. According to the authors, this combination improves gradient flow and mitigates vanishing gradients and dead neurons, yielding more stable optimization for the stiff, highly nonlinear SI1I2R dynamics.

The empirical results support this claim. When compared against a reference ODE solver, the boosted PINN’s predicted trajectories for all four compartments closely track the ground truth. The susceptible population declines exponentially as nodes become infected, the infected classes rise to a peak and then fall as recovery proceeds, and the recovered population grows and levels off. Quantitatively, the boosted framework reduced the mean squared error from 1.54 × 10⁻² to 6.22 × 10⁻³, an improvement of roughly 59.6 percent, cut the root mean squared error from 1.24 × 10⁻¹ to 7.89 × 10⁻², lowered the mean absolute deviation from 8.36 × 10⁻² to 2.58 × 10⁻², about a 69.1 percent improvement, and improved the Theil’s Inequality Coefficient from 2.49 × 10⁻³ to 1.58 × 10⁻³, roughly 36.5 percent, relative to the standard PINN.

Activation function choice proved decisive. The team compared Tanh, ReLU, SeLU, and the hybrid Swish–ReLU, examining training and validation loss curves for each. ReLU, though computationally efficient, can produce non-responsive neurons, while Tanh and Sigmoid offer smooth transitions but suffer from gradient disappearance. Swish–ReLU produced the smallest losses among all functions tested, and the authors identify it as the most stable and accurate option for this class of coupled nonlinear equations. The loss profiles also showed healthy learning behavior: the data loss dropped sharply in early epochs and stabilized, the physics residual loss fell rapidly and stayed near zero, and validation loss, after fluctuating in the first epochs, stabilized around 2,500 epochs and then declined toward zero, suggesting good generalization and limited overfitting.

Robustness was assessed across 20 independent training trials using MSE, TIC, and MAD. The MSE values ranged roughly from 10⁻¹ to 10⁻⁵, with the susceptible compartment centered near 10⁻² and the I1 compartment showing the largest variability, occasionally approaching 3 × 10⁻². The fully infected and recovered compartments were markedly more accurate, with errors generally below 10⁻³; the median MSE was approximately 2.8 × 10⁻⁴ for I2 and 1.0 × 10⁻³ for R, while medians for S and I1 were around 1.3 × 10⁻² and 1.4 × 10⁻² respectively. TIC values remained low, roughly between 10⁻³ and 10⁻¹, with the recovered compartment achieving the smallest values, and MAD values stayed within a controlled range without any systematic increase across trials, which the authors interpret as evidence of a repeatable and stable training process.

Beyond network performance, the study includes a grid-based sensitivity analysis of two key epidemiological parameters: the infection rate beta, varied from 0.01 to 0.2, and the progression rate alpha from I1 to I2, varied from 0.01 to 1.0, each discretized into 20 intervals to form a 20 × 20 grid. For each parameter pair, the model was solved numerically and the peak values of the I1, I2, and R compartments were recorded as heatmaps. The results show that higher infection rates produce larger I1 peaks, especially when alpha is low, while peaks in the fully infected and recovered compartments grow when both beta and alpha are large, identifying the most influential factors governing outbreak magnitude in the model.

The authors are careful about scope. The study relies on simulated data rather than real-world network traffic, assumes homogeneous mixing, and excludes stochastic effects and adaptive cybersecurity defenses. They also note that further evaluation with measures such as accuracy, precision, recall, or F1-score would better establish generalization. Within those limits, the work suggests that boosted PINNs, combining physics-based losses with hybrid activation functions, could become a resource-efficient tool for forecasting how computer viruses propagate through networked systems, and the authors point toward future work incorporating realistic network topologies, operational cybersecurity datasets, cyber-range experiments, stochastic components, and adaptive defense strategies.

Subject of Research: A boosted physics-informed neural network framework for modeling nonlinear computer-virus propagation with vertical transmission

Article Title: Boosted training strategy for physics-informed neural networks in modeling non-linear computer virus dynamics with vertical transmission

Article References: Ul Rahman, J., Shabeer, S., Mustafa, N., Akram, N., & Uwitije, R. (2026). Boosted training strategy for physics-informed neural networks in modeling non-linear computer virus dynamics with vertical transmission. Discover Artificial Intelligence, 6(1), Article 946. https://doi.org/10.1007/s44163-026-01945-2

Image Credits: AI Generated

DOI: 10.1007/s44163-026-01945-2

Keywords: physics-informed neural networks, computer virus dynamics, SI1I2R model, vertical transmission, cybersecurity, Swish-ReLU activation, e-epidemic modeling, sensitivity analysis, machine learning, ODE solvers, error metrics, Boosted