Epidemic parameters without training a network: one least-squares solve, errors down up to four orders of magnitude

Parameter estimation for epidemic compartment models can be cast as one closed-form least-squares solve instead of training a neural network per disease. Physics-informed regression previously estimated time derivatives from the data and evaluated the model residual at the data points; that derivative estimate became the accuracy bottleneck when sampling was coarse or noise large, while DINN-style methods trained a network per disease for 20 minutes to 91 hours. Across the eleven DINN benchmarks in noise-free and noisy settings, 22 model/noise combinations in total, physics-informed weak regression is the most accurate estimator in 15 and sixth-order finite-difference physics-informed regression in the remaining seven; where sampling resolves the dynamics they are one to four orders of magnitude more accurate than the reported DINN results, at about a millisecond per fit.

The new method integrates the model in time rather than differentiating the data, and the integrated equations stay linear in the parameters while the unknown initial state is solved jointly. The strong form approximates the time derivative of the states by finite differences, and higher orders reduce truncation bias but amplify noise; the weak form instead integrates the governing equations over time intervals, so the data themselves appear rather than their derivatives. The integrals are evaluated exactly from a cubic spline on the sampling grid with fourth-order error and weights that depend only on the time grid; on the COVID-19 SIDR model the noise-free weak-regression error is about one part in ten thousand, against about one percent for second-order finite differences.

Each estimate comes with intervals that hold for all parameters simultaneously, calibrated on samples the physics-based model generates itself. The least-squares covariance treats residuals as white noise and ignores truncation bias and residual correlation, so it is not trustworthy alone, and a model coefficient has no natural calibration set because each data set has exactly one true parameter vector. Drawing parameters from a prior box, simulating with the data's noise model and passing them through the same fixed estimator yields finite-sample, distribution-free simultaneous intervals; across 88 tested settings the nominal 90 percent intervals achieve 87 to 92 percent empirical coverage, narrow where estimates are accurate and wide where the data do not resolve the dynamics.

The method applies to compartment models whose right-hand side is linear in a coefficient vector, where coefficients are physical parameters or products and ratios of parameters; sampling must resolve the epidemic dynamics, and library terms carrying a parameter must not be swamped by noise in the largest compartment. The interval guarantee is marginal over the calibration population and assumes the compartment model itself is correct. On real data from the 1978 boarding-school influenza outbreak, a variable-projection variant gives calibrated 95 percent intervals that overlap the credible intervals of a Bayesian analysis; on the time-varying transmission benchmark a B-spline expansion keeps the problem linear and is more accurate than the physics-informed neural networks in three of four cases.

The coverage guarantee is marginal, holding on average over the calibration population rather than conditionally for a single observed data set, and it does not cover model misspecification or test cases far from the prior box. Errors-in-variables bias remains because the library is evaluated at noisy states, and the weak form averages out the random part of that error but not the bias. When there are more coefficients than physical parameters, a free fit ignores the constraint, which with few noisy samples can cost an order of magnitude in accuracy. On benchmarks such as Anthrax, whose outbreak lasts a few days but is sampled every 40 days, all estimators have median errors in the tens of percent.