RI-SINDy pulls Hill terms out of the regression: library conditioning drops 3–45× on oscillating gene networks
Synopsis
RI-SINDy removes known Hill-type regulatory terms from the sparse-regression library and solves each coefficient separately from a drift-balance condition, the near-zero time average of the derivative along a bounded trajectory, avoiding the ill-conditioning that arises when regulatory and polynomial terms are nearly collinear over an oscillation. On Hes1 experimental data it gives the lowest trajectory error among Traditional SINDy, SR3 and Nullcline Reconstruction; on synthetic Goodwin and NF-κB models it recovers the correct sparse structure, and in a Goodwin noise sweep it still finds all 6 active terms at 15% noise.
Interpretation
Saturating regulatory terms in gene networks no longer enter the same sparse regression as polynomial terms; they are solved separately, cutting the regression matrix condition number by 3–45×. Placing a Hill-type term in a shared candidate library made it nearly collinear with polynomial candidates over the range an oscillating trajectory covers, and ridge regularization, library normalization and orthogonalization only marginally improved conditioning without removing that structural collinearity. On a synthetic Goodwin model, appending two or more Hill columns to a polynomial library raised the raw-library condition ratio κ/κB to 38.9, 42.0 and 3.8 for d=1, 2 and 3; mean centering and variance normalization gave only marginal gains, and ridge regularization helped only slightly on highly correlated libraries.
The candidate library is split into a standard basis and regulatory terms, then updated alternately: standard-basis coefficients come from constrained least squares, and the regulatory coefficient is computed directly from the drift-balance condition. Drift balance uses the fact that the time-averaged derivative of a bounded trajectory approaches zero, so the regulatory term's mean contribution must offset the mean contribution of the remaining active terms, giving one scalar equation for one unknown regulatory coefficient. The method is tested on the Hes1 protein-mRNA network, the Goodwin limit cycle and the NF-κB signaling network against Traditional SINDy, SR3 and Nullcline Reconstruction SINDy, with a separate noise sweep against E-SINDy.
On real experimental data, RI-SINDy gives the lowest trajectory error on both Hes1 state variables among the compared methods, and its Monte Carlo uncertainty band contains most of the interpolated trajectory. SR3 assigned small nonzero weight to nearly every candidate term in both Hes1 equations, while Nullcline Reconstruction and Traditional SINDy returned large coefficients of mixed sign and none captured the observed oscillation. The Hes1 dataset has about 30 measurements over a 12-hour observation window and was interpolated with a sum of sines before fitting; Hill hyperparameters were set to literature values K0=2.7 and n=9, while SR3 searched to K0=2.41 and n=20.00, hitting its upper bound.
In the synthetic Goodwin noise sweep, RI-SINDy recovered all 6 active terms at each of 0, 5%, 10% and 15% noise and had the lowest relative L2 error at every level. In the same sweep, Traditional SINDy recovered 4, 3, 1 and 2 of 6 active terms and E-SINDy recovered 4, 2, 1 and 0; RI-SINDy's relative L2 errors were 0.200, 0.394, 0.274 and 0.675, against 0.633–2.525 for the other methods. The cost is that RI-SINDy retained 1, 3, 3 and 2 low-magnitude spurious terms at the four noise levels and was the only method with a spurious term at 0% noise.
Perspective
The demonstrated scope is low-dimensional systems: the Hes1 protein-mRNA network, the Goodwin limit cycle and the NF-κB signaling network, where NF-κB is a single-pulse transient rather than a sustained oscillation, only the IKK equation uses drift balance, and the boundary term contributes a relative error of about 7%. The intended readers are researchers and engineers working on equation identification in regulatory networks; the method requires the regulatory mechanism's functional form to be known and only one unknown regulatory coefficient per state equation.
Drift balance assumes a sufficiently long observation window and becomes less reliable when the trajectory is incompletely observed; if the regulatory term's own time-averaged contribution is near zero, estimation error is amplified by roughly 1/DR,j. When several regulatory coefficients are unknown in the same equation, one scalar constraint cannot determine them uniquely, and the authors list relaxing that restriction as the most immediate next step. The NF-κB results should be read as a test on a decaying transient with a boundary term, not as a limit-cycle test.