AlphaNet7: a multi-head physics-informed neural network for latent parameter discovery in variable-order fractional systems

Abstract Solving high-dimensional inverse problems in variable-order fractional dynamical systems from sparse observations remains a major challenge in scientific machine learning. Standard Physics-Informed Neural Networks (PINNs) rely on soft residual penalties, which frequently induce gradient pathologies or encounter conservation discrepancies when applied to non-local fractional operators. To address this limitation, we propose AlphaNet7, a constrained multi-head PINN architecture for the simultaneous inverse identification of compartment-specific, time-varying fractional orders and transmission dynamics. The proposed architecture combines a shared feature-extraction backbone with eight independent prediction heads, jointly inferring multiple latent quantities while implicitly integrating the governing phenomenological equations through a custom autograd-compatible piecewise differentiable explicit fractional forward solver. AlphaNet7 is applied to a seven-compartment nonlinear epidemiological system governed by the non-singular Caputo-Fabrizio fractional operator. From two macroscopic surveillance streams, the network estimates a regularization-guided, constraint-dependent set of seven phenomenological effective fractional orders rather than genuine biological memory parameters. In the reference run, inferred compartment orders span approximately 0.688 to 0.864 (mean 0.753), whereas a single-head global-order model achieves identical reconstruction error (RMSE = 0.0028) and removing the diversity penalty contracts compartment means to between 0.740 and 0.774 (RMSE = 0.0021). Under regularization-guided optimization, the multi-head model captures nonlinear transitions in exposure and recovery alongside quasi-stationary institutional plateaus; this alignment reflects post hoc temporal coincidence with policy shifts rather than causality. Furthermore, the inferred regime yields a modest reduction of roughly 2% to 3% (1.38 to 1.35) in the definition-driven, phenomenological fractional reproduction metric. Because the explicit solver numerically constrains trajectories to the discretized governing dynamics, AlphaNet7 outperforms fixed-order baselines in calibration and out-of-sample forecasting, reducing forecast error by 30.2% to 63.1% across rolling hold-out splits under unified normalization. This provides a prior-constrained framework for effective parameter estimation in variable-order fractional systems.