Multi-stage physics-informed neural networks for JAK--STAT5 signaling and ultradian insulin--glucose dynamics: latent-species identifiability and suppression of parameter-induced divergence

Coupled diffusion--reaction partial differential equations (PDEs) describe biochemical network dynamics but are difficult to solve for realistic multi-species systems without combining mechanism and data. We present a multi-stage physics-informed neural network (PINN) for multi-species diffusion--reaction PDEs and apply it to two ordinary-differential-equation (ODE) reference systems: the Boehm et al. JAK--STAT5 signaling pathway and the Sturis ultradian insulin--glucose model. For STAT5 we pose a latent-species emph{identifiability} test: given sparse observations of eight species, a ten-species model that retains two deliberately withheld but mechanistically standard components---an active receptor--JAK complex and the SOCS negative-feedback inhibitor---recovers the reference trajectory and reduces mean root-mean-square error 3.1-fold relative to an eight-species model that omits them, whereas a PDE-only solution without data anchoring diverges. Because the reference is itself ODE-generated, this demonstrates identifiability against synthetic data, not the discovery of new biology. For the insulin--glucose model the same framework reproduces the ${sim}120$-minute oscillation to 1.0% mean relative error as a benchmark on a stiff, multi-timescale oscillator; its spatial dimension is treated as a numerical construct, not a physical transport setting. A Lyapunov analysis of the STAT5 ODE returns a maximal exponent statistically indistinguishable from zero ($lambda_{max} approx 3.61 times 10^{-5};text{min}^{-1}$, 5/8 trials positive; Lyapunov time ${sim}1.9times10^{4}$~min, far exceeding the 240--720~min horizon), so the system is effectively non-chaotic and the relevant instability is a bounded, parameter-induced trajectory divergence. Anchoring the solution to baseline data suppresses this divergence, with the reduction growing monotonically with sampling density---from ${sim}15$--$19%$ at eight time points to ${sim}88$--$97%$ at sixty-four, depending on perturbation magnitude. The framework thus offers a data-anchored route to latent-species identifiability and divergence suppression in biochemical ODE/PDE systems,demonstrated here against synthetic reference data.

Deng, J., Zhang, X., Zhang, X., Yang, X.

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