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Conformalized Physics-Informed Weak Regression for Parameter Estimation in Epidemic Models

Allan Peter Engsig-Karup

arXiv:2610.19058v1stat.MLcs.LGmath-phmath.MP

TL;DR

Estimating compartment-model rates from time series is an inverse problem, and those coefficients determine reproduction numbers, peak sizes, and intervention effects. The paper uses least-squares physics-informed regression, including a time-integrated weak form, with model-generated split-conformal calibration for simultaneous parameter intervals. On eleven DINN benchmarks, PIWR or PIR-FD6 improves reported errors by up to four orders of magnitude wherever sampling resolves the dynamics, and 90% intervals retain coverage in all 88 tested settings.

  • Problem

    Estimating compartment-model rates from time series is an inverse problem; these coefficients determine reproduction numbers, peak sizes, and intervention effects.

    Source: §1 · p. 2

  • Method

    The paper uses least-squares physics-informed regression, including PIWR's time-integrated weak form, and model-generated split-conformal calibration for simultaneous parameter intervals.

    Source: §6 · pp. 18–19

  • Results

    On eleven DINN benchmarks, PIWR or PIR-FD6 improves reported errors by up to four orders of magnitude wherever sampling resolves the dynamics, while 90% intervals retain coverage in all 88 tested settings.

    Source: §6 · pp. 18–19

  • Takeaways & Limitations

    Intervals narrow where estimates are accurate and widen where data do not support them, without access to true parameters.

    Source: §6 · p. 19

    The guarantee is marginal over the calibration population and assumes the model is correct; misspecified models and test cases far from the prior box are not covered.

    Source: §5.10 · p. 18

Abstract

from arXiv · show

Compartment models of infectious diseases are often linear in coefficients that are physical parameters or products or ratios of them. We exploit this with closed-form least squares and equip the estimates with distribution-free uncertainty. Physics-informed regression (PIR) fits the parameters to finite-difference derivatives of the data; we introduce physics-informed weak regression (PIWR), which integrates the model in time instead, needs no derivatives of the data, estimates the initial state jointly and is high-order accurate. To quantify uncertainty we wrap both estimators in split conformal prediction, with a calibration population generated by the physics-based model itself, giving simultaneous intervals for all parameters with a finite-sample coverage guarantee. A residual-adaptive factor, computed without ground truth, scales the intervals to each data set (PIWR-RACC). On the eleven benchmark models of disease-informed neural networks (DINNs), PIWR or PIR-FD6 is one to four orders of magnitude more accurate than the reported DINN results wherever the sampling resolves the dynamics, in about a millisecond per fit instead of minutes to hours of training, and the 90% intervals achieve 87-92% coverage in all 88 settings. On the S3I3R model of the original PIR study, PIWR lowers the reported PIR errors by about three orders of magnitude. On real data from the 1978 boarding-school influenza outbreak, a variable-projection variant gives calibrated 95% intervals that overlap those of a Bayesian analysis. On a time-varying transmission benchmark, a B-spline expansion of beta(t) keeps PIR and PIWR linear; they are more accurate than physics-informed neural networks in three of four cases and comparable in the noisiest, with simultaneous bands for beta(t). The complete conformal calibration takes less than two minutes for every benchmark.

1 Introduction

Parameter estimation in compartment models is often a linear least-squares problem, but derivative-based fits can be fragile and point estimates alone do not express their uncertainty. The paper develops weak-form regression and model-generated conformal calibration to address these issues.

1 Introduction

  • PIR replaces neural-network training with a closed-form least-squares solve, but its accuracy depends on estimating derivatives from the data.The derivative estimate is especially problematic for coarse or noisy data.Source: §1 · p. 2
  • Compartment-model coefficients encode physical rates, but estimates can be affected by noise, discretization, identifiability, and mismatch between a model and a real epidemic.The authors argue that uncertainty matters because estimates such as reproduction numbers inform interventions.Source: §1 · p. 2

1.1 Contributions

  • PIWR integrates the model in time, avoiding data derivatives, jointly estimating the initial state, and achieving fourth-order accuracy with cubic-spline quadrature.The integrated equations remain linear in the coefficients.Source: §1.1 · p. 3
  • Split conformal calibration uses simulated epidemics with known parameters to provide simultaneous intervals for all parameters with a finite-sample, distribution-free coverage guarantee.The least-squares covariance shapes the intervals, while calibration sets their common scale.Source: §1.1 · p. 3
  • Residual-adaptive calibration scales intervals using a regression residual available without ground truth, widening them for difficult data and narrowing them for easier data while retaining the coverage guarantee.The paper evaluates the methods on eleven DINN benchmarks and additional S3I3R, real-outbreak, and time-varying-transmission cases.Source: §1.1 · p. 3

2 Regression framework: parameter-linear epidemic models

The framework exploits compartment models whose equations are linear in coefficients, even when those coefficients are products of physical parameters. Regression estimates the coefficients, which are then mapped back to the physical parameters.

2 Regression framework: parameter-linear epidemic models

  • The framework assumes measured states are a noisy version of a solution to an initial-value problem, an assumption that may fail for real data through model error.The model includes an unknown initial state along with physical parameters.Source: §2 · p. 3
  • When a model is linear in coefficients that are parameters or parameter products, least squares estimates the coefficients and a left inverse maps them back to physical parameters.Known model terms can enter as offsets and be subtracted from the data side of the regression.Source: §2 · p. 4
  • Parameters that enter only as a product cannot be separated by any method from the corresponding single library column, so one parameter must be fixed.For p > k, the coefficient vector also satisfies p − k constraints.Source: §2 · p. 4
  • In the seasonal measles example, expanding the transmission term makes the equations linear in coefficients such as β0β1 and µσ, with physical parameters recovered through a map.The seasonal amplitude is recovered as the coefficient ratio β1 = ϑ3/ϑ2.Source: §2 · p. 4
  • Table 1 summarizes the eleven benchmark models’ state and parameter counts, fixed parameters, sampling grids, and reported DINN training times.The table also lists coefficient-vector lengths and the corresponding DINN tables.Source: Table 1 · p. 5

3 Physics-informed regression

PIR and PIWR formulate parameter estimation as linear least squares for compartment models. PIR uses finite-difference derivatives, while PIWR integrates the equations and fits the initial state jointly; the resulting coefficient estimates are mapped to physical parameters.

3.1 Strong form: physics-informed regression (PIR)

  • PIR stacks finite-difference estimates of state derivatives and model-library evaluations into an overdetermined least-squares system.Higher-order differences reduce truncation error but amplify derivative noise: for FD2, FD4, and FD6, noise standard deviations are 0.71σ/∆t, 0.95σ/∆t, and 1.08σ/∆t.Source: §3.1 · p. 5
  • PIR can scale state equations so compartments of different magnitudes weigh comparably, and scale columns to improve conditioning without changing the minimizer.Whether to use row scaling is selected per model and estimator on a tuning set.Source: §3.1 · p. 5

3.2 Weak form: physics-informed weak regression (PIWR)

  • PIWR integrates the model and regresses sampled states rather than their derivatives, keeping the relation linear in the coefficients.It jointly estimates the initial state so noise in the first sample is not propagated into every equation.Source: §3.2 · p. 6
  • Cubic-spline quadrature integrates the interpolated library with O(∆t^4) error for smooth integrands, compared with O(∆t^2) for the trapezoidal rule.The weights depend only on the time grid and are computed once.Source: §3.2 · p. 6
  • PIWR solves a least-squares system for coefficients and initial state using orthogonal factorization, avoiding normal equations that square the condition number.When the sampled trajectory identifies the parameters, the relevant centered system is nonsingular.Source: §3.2 · pp. 6–7
  • PIWR uses all samples, averages noise in the integrals, and needs to resolve the area under a sharp outbreak rather than its slope, but still depends on resolving the sampled library.It also inherits errors-in-variables bias because the library is evaluated at noisy states.Source: §3.2 · p. 7

3.3 From coefficients to parameters

  • The physical parameters are recovered through a map from the fitted coefficients, with uncertainty propagated approximately by the delta method using the map’s Jacobian.The ordinary least-squares covariance treats residuals as white noise and ignores truncation bias and residual correlation, so it is not trustworthy alone.Source: §3.3 · p. 7

4 Conformal prediction with physics-based models

The section uses simulated epidemics with known parameters to calibrate simultaneous intervals for all model parameters, then adapts interval widths to each data set using its regression residual. The finite-sample guarantee depends on exchangeability with the calibration population and on the data following the model.

4.1 A calibration population from the model

  • Calibration examples are simulated from a parameter prior box with the same initial condition, time grid, and noise model, then passed through the fixed PIR or PIWR estimator.Tuning is performed on separate draws so calibration scores remain exchangeable with scores for new problems from the same population.Source: §4.1 · p. 7

4.2 Simultaneous intervals

  • Under exchangeability, the intervals cover all parameters simultaneously with finite-sample coverage; the guarantee is marginal over the parameter population and does not apply when data come from a different model.Any estimator bias or noise distribution is allowed, though a poor interval shape can increase width.Source: §4.2 · p. 8

4.3 Residual-adaptive calibration and PIWR-RACC

  • The residual-adaptive factor scales scores and intervals using the data-only regression residual, widening intervals for difficult fits and narrowing them for easier ones without changing the coverage guarantee.The reference residual is the median across 200 tuning draws.Source: §4.3 · p. 8

5 Results

The results evaluate PIWR and PIR across benchmark epidemic models and time-varying transmission, assessing parameter accuracy and conformal interval or band coverage. They also examine sampling resolution, prior knowledge, computational cost, and limits on accuracy and coverage.

5.1 Setup

  • The eleven DINN benchmarks were regenerated on the original initial conditions, time grids, and parameter values, with noise-free and 1 % noise settings and 999 calibration problems.Coverage was evaluated on 2,000 independent test problems; estimator tuning used separate draws.Source: §5.1 · pp. 8–9

5.2 Worked example: COVID-19

  • For SIDR without noise, PIWR reached 0.0001 % error, four orders of magnitude below DINN, in a least-squares solve of less than a millisecond.With 1 % noise, PIWR error was 0.12–0.49 % and its intervals were narrower than PIR-FD6 intervals (median ±1.9 % with PIWR-RACC against ±4.5 % for PIR-FD6).Source: Table 2 · p. 9, §5.2 · p. 9, Figure 2 · p. 9

5.3 Accuracy against the sampling interval

  • At the DINN spacing, PIWR was 30 times more accurate than PIR-FD6 for COVID-19 and achieved 9.4 % error for Measles, where no finite-difference estimator was better than 33 %.PIWR reached 0.1 % Measles error when the sampling interval was halved, and 10^-3 % when halved twice.Source: §5.3 · pp. 9–10, Figure 3 · p. 10

5.4 Bechmarks across all eleven models

  • Without noise, PIWR or PIR-FD6 improves reported DINN errors by one to four orders of magnitude for COVID-19, Ebola, Dengue, Pneumonia, Polio, and Tuberculosis; HIV and Zika are comparable, while Smallpox and Measles are less accurate.With noise, reported errors are Pneumonia 11% against 13–57%, Ebola 25% against 37–40%, Measles 28% against 84–88%, and HIV 22% against 35–51%.Source: §5.4 · p. 10

5.5 Uncertainty quantification

  • Across 88 settings, empirical coverage was 87.6–92.0 % for plain intervals and 86.7–92.5 % for adaptive intervals, with means of 90.0 % and 89.9 %.With 1 % noise, PIWR intervals were narrower than PIR-FD6 intervals for nine of eleven models; the residual-adaptive factor narrowed intervals in some heterogeneous cases and widened them slightly when the literature case was typical.Source: §5.5 · p. 11, Table 4 · p. 12, Figure 5 · p. 12

5.7 Prior knowledge in the conformal calibration

  • Prior knowledge narrowed some intervals only when its uncertainty was represented in calibration; calibration assuming exact recovery rates covered just 2.4 % of S3I3R tests with rates varying by up to ±20 %.A ±20 % calibration restored 89.9 % coverage; for influenza, drawing γ from [0.4, 0.6] in calibration yielded 95.2 % coverage when test γ varied across that range.Source: §5.7 · pp. 14–15

5.8 Time-varying transmission: comparison with PINNs

  • The time-varying transmission benchmark uses daily infected counts over 90 or 120 days to compare methods estimating β(t) or Rt, including a constant-rate case, two-wave cases, and a noisy case.The reference β(t) for Cases 2–4 was digitized from published figures and adjusted to reproduce the published infected-count curves.Source: §5.8 · pp. 15–16
  • After day 20, PIWR errors are 5.0 · 10^-3 for β(t) and 2.7 · 10^-3 for Rt, versus 1.6 · 10^-2 and 6.7 · 10^-2 for the better PINN (Millevoi et al., 2024).In Cases 1–3, PIR and PIWR are generally more accurate than both PINN variants, mostly by one to two orders of magnitude; in Case 4, PIR-FD6 has Rt errors of 0.17 over the whole period and 0.18 after day 20, versus 0.47–0.50 and 0.31–0.33 for PINNs. Fits take milliseconds rather than 12–37 minutes of training.Source: §5.8 · p. 16, Table 9 · p. 17
  • Conformal bands cover 89.2–91.9% of test curves for both estimators, while Case 4 coverage is 88.2–89.8%.For Case 3, PIWR-RACC reduces the median half-width from 28% to 3.0% by dividing scores by the residual-adaptive factor; the bands contain the published curves in all 200 noise realizations.Source: §5.8 · pp. 16–17, Figure 7 · p. 18

5.9 Computational cost

  • PIR fits take 0.14–0.5 ms and PIWR fits 0.5–1.6 ms, while the full conformal pipeline takes 19–108 s versus 20 minutes to 91 hours for reported DINN training.The runtime comparison is indicative because the hardware differs; the reported gap is three to five orders of magnitude.Source: §5.9 · p. 17

5.10 Limitations

  • PIR and PIWR require samples that resolve the dynamics, and parameter-carrying library terms must not be overwhelmed by noise in the largest compartment.The latter constraint limits, for example, hospital and funeral transmission-rate estimates in Ebola.Source: §5.10 · p. 17
  • Noisy regressors can bias nonlinear library terms and shrink least-squares coefficients; weak-form averaging removes random error but not bias, which can dominate sparse noisy counts.Conformal calibration retains coverage despite this effect, at the cost of wider intervals; smoothing, bias correction, or total least squares could reduce it.Source: §5.10 · p. 17
  • Free fits can lose an order of magnitude in accuracy when coefficients are constrained, while variable projection restores the constraint for nonlinearities in one or two parameters; the guarantee assumes a correct model and calibration-population coverage.Sparse-count likelihood fits remain more efficient than least squares, and cases far from the prior box or under model misspecification are not covered.Source: §5.10 · p. 18

6 Conclusions

The conclusions present physics-informed regression as a fast alternative to neural-network training for coefficient-linear epidemic models, with PIWR avoiding derivative estimates. They also report calibrated uncertainty for constant and time-varying parameters, while limiting the practical recommendation to data that resolve epidemic dynamics.

6 Conclusions

  • Up to four orders of magnitude: PIWR or PIR-FD6 improves on reported DINN errors wherever sampling resolves the dynamics, at milliseconds instead of hours.The authors say PIWR-VP was essential with only 15 boarding-school outbreak samples, and that alternative methods such as Bayesian inference remain valuable for validation.Source: §6 · p. 18
  • Three of four time-varying benchmark cases favor PIR and PIWR over PINNs; in the noisiest case they are at least comparable, with millisecond runtimes instead of tens of minutes.B-spline expansion keeps the model linear in coefficients, while conformal bands for beta(t) and Rt widen under 40% errors on every count.Source: §6 · p. 18
  • 90% simultaneous intervals retained coverage in all 88 tested settings, and PIWR-RACC adapts interval width to each dataset without coverage cost.In time-varying Case 3, PIWR-RACC narrows bands by a factor of nine; approximate prior knowledge must be represented in calibration, or coverage is lost.Source: §6 · p. 19

Declaration

The author reports no potential conflicts of interest and describes limited use of generative AI tools during preparation.

Declaration

  • The author declares no potential conflicts of interest concerning the investigation, authorship, or publication.Source: Declaration · p. 19
  • Generative AI tools were used to review Python code, set up systematic numerical experiments, and improve language clarity; the author takes responsibility for the scientific content.The author states that the original ideas came from human thoughts and creativity.Source: Declaration · p. 19

Code and hardware environment

The implementation uses standard scientific Python tools and dense double-precision least-squares solves, without deep-learning frameworks, automatic differentiation, or GPUs.

Code and hardware environment

Data

The data section identifies the sources and generation procedures for the benchmarks, outbreak data, and time-varying cases, then specifies the common model and table conventions for the DINN models.

Data

A The eleven DINN epidemic models

  • The eleven DINN models are specified through their state equations, physical parameters, linear coefficients, inverse parameter map, known constants, and initial condition.When coefficient count exceeds parameter count, the resulting constraints are listed but not imposed.Source: A The eleven DINN epidemic models · p. 19
  • The tables report PIWR with PIWR-RACC 90% simultaneous intervals for noise-free and 1% noise data, alongside DINN estimates.A dagger marks absolute errors for parameters whose true value is zero.Source: A The eleven DINN epidemic models · p. 20

A.1 COVID-19: SIDR model

This section specifies the SIDR model’s state, parameter mapping, and sampling setup. Table 10 presents PIWR estimates and PIWR-RACC 90% simultaneous intervals alongside DINN values.

A.1 COVID-19: SIDR model

  • The SIDR state is (S, I, D, R), with known population N = 59 · 10^6.The section also gives the model’s parameter mapping and initial condition.Source: §A.1 · p. 20
  • The model library is given in matrix form, and the data comprise 100 samples over 500 days.Source: §A.1 · p. 20
  • Table 10 presents PIWR estimates and PIWR-RACC 90% simultaneous intervals alongside DINN values from Shaier et al. (2022).The supplied passage identifies the table’s contents but does not provide its numerical entries.Source: Table 10 · p. 20

A.2 HIV

These benchmark subsections specify the state variables, coefficient mappings, and data settings for six disease models. They also record model-specific constraints and fixed quantities relevant to parameter estimation.

A.2 HIV

  • The HIV model has three state variables, eight listed coefficient terms, initial state (1000, 10, 0.01), and one constraint, ϑ4 = (ϑ2 + µT)ϑ8.The known natural death rate µT and burst factor N are fixed; the data use 50 samples over 20 days.Source: §A.2 · p. 20

A.3 Smallpox

  • For smallpox, βφSI and βSI are rewritten as 10γSI/N and 10γS2/N, respectively, yielding a linear coefficient representation.The model has eight states, ten listed coefficients, two constraints, and 50 samples over 100 days.Source: §A.3 · p. 21

A.4 Tuberculosis: SLIT model

  • The tuberculosis SLIT model estimates eight parameters, while the contact rate is fixed because it enters only through βc and β′c.The model uses 50 samples over 40 days, and d = 0 is reported with absolute error.Source: §A.4 · p. 22

A.5 Pneumonia: SVCIR model

  • The pneumonia SVCIR model uses a 15-term coefficient mapping and one constraint, ϑ10 = ϑ8ϑ9/ϑ6.Its data comprise 50 samples over 100 days; the DINN table parameters k and τ do not occur in the model.Source: §A.5 · p. 22

A.6 Ebola: SEIHFR model

  • In the Ebola SEIHFR model, the known probabilities θ1, δ1, and δ2 enter as rate factors and are therefore fixed.The model has six states and uses 100 samples over 100 days.Source: §A.6 · p. 23

A.7 Dengue: host–vector model

  • For dengue, the host–vector model has 11 listed coefficients and one constraint, ϑ4 = ϑ3ϑ5; the biting rate is fixed because it enters only through ab and ac.The DINN table’s λv does not occur in the model, and the data use 50 samples over 200 days.Source: §A.7 · pp. 23–24

A.8 Anthrax: SIAC model

These four benchmark subsections document model-specific coefficient representations, fixed quantities, and data settings. They also note discrepancies between one benchmark’s data-generation setup and its printed reference values.

A.8 Anthrax: SIAC model

  • The Anthrax SIAC benchmark’s DINN data file differs from the accurate solution by up to 24%.The generator uses δ = 1/20 and β = 1/500, differing from the paper table values 1/64 and 0.02.Source: §A.8 · p. 25

A.9 Polio: age-structured SIR model

  • For polio, the child and adult group sizes are fixed by the initial condition, and the cross rates βac = βca = 0 are reported with absolute errors.The model has eight listed coefficients and uses 50 samples over 40 days.Source: §A.9 · p. 25

A.10 Measles: seasonally forced SEI model

  • The measles model estimates β0 even though the DINN benchmark treats β0 as known, and its known seasonal forcing makes the library explicitly time-dependent.The forcing has period one year; the data comprise 50 samples over 10 years.Source: §A.10 · p. 26

A.11 Zika: host–vector model

  • In the Zika host–vector model, the known biting rate is fixed because it enters only through ab and ac, and three constraints link its coefficients.The model has nine states and uses 50 samples over 30 days.Source: §A.11 · pp. 26–27

B Further benchmark models

These further benchmarks cover an S3I3R model with unknown rates and an SIR model where only infection counts are observed. The latter uses a data-derived quadrature quantity in its regression library.

B.1 S3I3R model

  • The S3I3R setting estimates all seven rates, whereas the Nielsen et al. (2025) setting assumes the recovery rates known and moves their terms into the offset.The data use 29 daily samples over 28 days.Source: §B.1 · p. 28

B.2 SIR model with only the infected observed

  • For the SIR benchmark, only I is observed, and its closed regression equation uses the scalar state x = I.The library depends on the trajectory’s past through C, computed from the data by cubic-spline quadrature; the dataset is the 1978 boarding-school influenza outbreak.Source: §B.2 · pp. 28–29
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