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Learning and Forecasting Multiscale Spatiotemporal Epidemic Dynamics with Lagrangian-Eulerian Graph Neural ODEs

Jinyi Duan, Qian Li, Biao Tang

arXiv:2610.21223v1cs.LG

TL;DR

Long-range epidemic forecasting must extrapolate complex dynamics from partial observations, while hybrid models often leave unclear how observations constrain epidemic processes and how learned interactions change state. LE-EpiGNODE uses process-specific contact and mobility operators in structured dynamics; it achieved the lowest WMAPE at the longest horizon across three real-world datasets.

  • Problem

    Accurate medium- to long-range epidemic prediction is difficult because trajectories emerge from high-dimensional, nonstationary, multiscale dynamics.

    Source: Introduction · p. 1

  • Method

    LE-EpiGNODE maps age-contact information to Lagrangian infection generation and mobility information to Eulerian conservative population transport in a structured UDE.

    Source: Introduction · p. 3

  • Results

    At the longest prediction horizon, LE-EpiGNODE achieved the lowest WMAPE in all three real-world datasets, with gains generally becoming more pronounced at longer horizons.

    Source: Discussion · pp. 11, 13

  • Takeaways & Limitations

    Mechanism-defined message semantics and state actions make learned interaction strengths and induced state fluxes directly testable against synthetic ground truth.

    Source: Discussion · p. 13

    The auxiliary mobility network is pretrained on historical OD observations and frozen during epidemic-model training, which may limit adaptation to abrupt mobility changes.

    Source: Discussion · p. 13

Abstract

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Accurate medium- to long-range epidemic forecasting requires extrapolating high-dimensional, heterogeneous and nonstationary dynamics from partial observations. Yet hybrid models often leave implicit which observations inform which epidemic processes and how learned graph interactions modify system state. We introduce LE-EpiGNODE, which replaces generic message passing with process-specific, mechanism-guided neural operators: age-contact information drives Lagrangian infection generation, whereas mobility drives conservative Eulerian population transport within a structure-preserving universal differential equation. We evaluated LE-EpiGNODE across 14 controlled synthetic mechanisms spanning time-modulated, state-dependent, nonlinear and coupled dynamics, three complementary COVID-19 settings, 21 comparator models and five targeted ablations. Forecasting gains tended to become more pronounced at longer horizons; at the longest horizon, LE-EpiGNODE achieved the lowest WMAPE in all three real-world datasets, reducing error by 17.7-40.0% relative to the best baseline. Controlled ablations linked these gains to the Lagrangian-Eulerian decomposition and process-aligned priors, while synthetic ground truth showed that learned operators and induced state fluxes track known generating dynamics. These results support mechanism-organized graph neural dynamics as a route to reliable long-horizon extrapolation with directly testable, process-level mechanistic interpretation beyond trajectory accuracy.

Introduction

Epidemic forecasting must extrapolate coupled, heterogeneous dynamics from partial observations, but existing models often leave process-specific information use and message-driven state changes implicit. LE-EpiGNODE addresses this with contact-informed infection generation and mobility-informed conservative transport operators in structured dynamics.

Introduction

  • At longer horizons, forecasts must extrapolate high-dimensional, nonstationary, multiscale epidemic dynamics from partial observations rather than simply extend observed trajectories.Incidence, contact, and mobility data provide complementary but incomplete views of latent states and interactions.Source: Introduction · pp. 1–2
  • Mechanistic models organize contact and mobility [6] [7], while graph and hybrid approaches combine learned relational structure with epidemic dynamics at parameter, graph, and continuous-dynamics levels [11] [12] [17].Mechanistic interaction functions become difficult to specify as heterogeneity, dimensionality, and temporal variation increase.Source: Introduction · p. 2
  • A central gap is that heterogeneous observations and graph messages often lack explicit alignment with the physical processes they constrain and the state changes they may induce.Contact-driven infection generation creates compartmental transitions, whereas mobility-driven transport redistributes existing states under conservation.Source: Introduction · p. 2
  • Graph neural learning and structured governing dynamics jointly address unresolved heterogeneous interactions while defining their physical meaning and admissible state action.The framework responds to complementary needs on the graph-learning and mechanistic-modeling sides.Source: Introduction · pp. 2–3, Figure 1 · p. 3
  • LE-EpiGNODE assigns age-contact information to Lagrangian infection generation and mobility information to Eulerian population transport within structured governing dynamics.Neural modules learn state- and time-dependent interaction strengths, while epidemic mechanisms specify message semantics and admissible state updates.Source: Introduction · p. 3, Figure 1 · p. 3

Results

Across synthetic and real-world COVID-19 settings, LE-EpiGNODE’s forecasting advantage tended to increase at longer horizons, and it achieved the lowest WMAPE at the longest horizon in all three real-world datasets. Ablations linked forecasting gains to process decomposition and aligned priors, while synthetic ground truth tested dynamical fidelity and real-world maps exposed process-level information patterns.

Results

  • The evaluation covered 14 synthetic generation conditions, three complementary COVID-19 settings, 21 comparator models, and five targeted ablations.Synthetic mechanisms varied temporal modulation, state feedback, nonlinear responses, and coupling; forecasts were evaluated at 1-, 4-, 12-, and 20-week horizons.Source: Results · p. 4, Synthetic epidemic forecasting · p. 4

Synthetic epidemic forecasting.

  • LE-EpiGNODE led in 10 of 12 synthetic setting–horizon combinations; at 20 weeks, WMAPE fell 65.7% vs SIR (21.11% vs 61.63%) in Eulerian and 47.6% vs MepoSIR (13.36% vs 25.52%) in multiscale settings.Its relative advantage was larger at medium and long horizons in Eulerian and multiscale settings, though several baselines were better at the shortest multiscale horizon.Source: Synthetic epidemic forecasting · pp. 4–5, Table 1 · p. 7

Real-world epidemic forecasting.

  • At the longest horizon, WMAPE fell 25.6% vs MCM-SEIR in Ontario (16.20% vs 21.76%), 40.0% vs MepoGNN in Hubei (15.87% vs 26.45%), and 17.7% vs MCM-SEIR in Spain (20.43% vs 24.81%).LE-EpiGNODE ranked first at every evaluated horizon in Ontario and Hubei, and at 49 and 93 days in Spain.Source: Real-world epidemic forecasting · p. 5, Table 1 · p. 7

Synthetic-system ablations.

  • At 20 weeks, removing LE structure raised WMAPE from 13.36% to 58.26% in multiscale; removing GL or GE raised it from 12.45% to 50.13% in Lagrangian and 21.11% to 78.27% in Eulerian settings.The particularly large Eulerian deterioration is consistent with state-dependent transport and CRGM’s temporal modulation; the multiscale gap supports separating infection generation from transport.Source: Synthetic-system ablations · p. 7, Table 2 · p. 9
  • Removing the age-contact prior raised 20-week WMAPE from 12.45% to 72.08% in Lagrangian, while removing OD-informed mobility pretraining raised it from 21.11% to 44.57% in Eulerian settings.The same variants increased multiscale errors, supporting process-matched assignment of contact and OD priors to their corresponding pathways.Source: Synthetic-system ablations · pp. 7–8, Table 2 · p. 9

Real-world ablations.

  • At the longest horizon, removing LE structure raised WMAPE from 16.20% to 34.78% in Ontario, 15.87% to 45.45% in Hubei, and 20.43% to 55.43% in Spain.Removing GL raised Ontario WMAPE to 58.20% and Spain WMAPE to 35.86%; removing GE raised Hubei WMAPE to 26.06% and Spain WMAPE to 23.44%. Removing the age-contact prior raised Ontario WMAPE to 28.08%, and removing mobility pretraining raised Hubei WMAPE to 32.22%.Source: Real-world ablations · pp. 8–9, Table 2 · p. 9

Dynamical fidelity in synthetic systems.

  • Without the age-contact prior, LIF-SRCC stayed 0.971 while ΦL-WMAPE rose from 25.89% to 196.68% and ΦL-PCC fell from 0.959 to 0.074, separating accumulated ranking from instantaneous flux fidelity.Across synthetic settings, learned operators and fluxes showed strong agreement with generating dynamics: Lagrangian FL-PCC, ΦL-PCC, and LIF-SRCC were 0.980, 0.959, and 0.971; Eulerian values were 0.996, 0.981, and 0.923. In multiscale, Lagrangian FL-PCC and ΦL-PCC were 0.936 and 0.920, Eulerian FE-PCC and ΦE-PCC were 0.999 and 0.990, and LIF-SRCC and EIF-SRCC were 0.835 and 0.998.Source: Dynamical fidelity in synthetic systems · p. 10, Figure 4a · p. 10

Process-level interpretation in real epidemics.

  • Real-world LIF/EIF maps expose age-group and regional information propagation—including heterogeneous age patterns in Ontario and Spain and Wuhan-centered fluxes in Hubei—not contact rates or causal transmission pathways.In Spain, central regions produced stronger outward messages; the maps are process-level explanations of forecast information, not causal transmission pathways.Source: Process-level interpretation in real… · p. 11

Discussion

LE-EpiGNODE’s forecasting advantage tends to grow at longer horizons, while its process-organized operators make learned dynamics testable in synthetic systems and interpretable in real-world forecasts. The discussion also identifies limits in identifiability, mobility adaptation, and evaluation scope.

Discussion

  • LE-EpiGNODE aligns age-contact information with Lagrangian infection generation and mobility with Eulerian population transport, while governing dynamics constrain learned state transformations.This mechanism-organized operator design targets long-horizon extrapolation and process-level interpretation beyond trajectory accuracy.Source: Discussion · pp. 11, 13, 14
  • At the longest horizon, LE-EpiGNODE achieved the lowest WMAPE in all three real-world datasets, with advantages generally more consistent as horizons lengthened.The relative pattern was not uniform: Ontario’s WMAPE reduction was larger at intermediate horizons, and LE-EpiGNODE was slightly less accurate than MepoSIR at 20 weeks in the Lagrangian synthetic setting.Source: Discussion · pp. 11, 13
  • Ablations support separating infection generation from transport and aligning each information source with its process; cumulative interaction rankings can remain stable despite sharply worse instantaneous flux agreement.The discrepancy is especially relevant to time-modulated or state-dependent mechanisms, where evolving operator behavior matters beyond integrated ordering.Source: Discussion · p. 13
  • The Fb →Φ →LIF/EIF hierarchy links learned interactions, their state-flux effects, and cumulative organization, enabling synthetic-ground-truth validation and process-level interpretation in real epidemics.In real data, these quantities support model interpretation rather than causal mechanism estimation.Source: Discussion · pp. 13–14
  • Latent epidemic states and learned process fluxes may be only partially identifiable from reported incidence, particularly with sparse, delayed, or reporting-affected observations.Additional epidemiological and behavioral measurements could provide stronger constraints on hidden dynamics.Source: Discussion · p. 13
  • Frozen parameters in the mobility network pretrained on historical OD observations may limit adaptation to abrupt mobility changes; evaluation also remains limited to stylized mechanisms and three COVID-19 settings.The authors call for broader validation across pathogens, intervention regimes, spatial scales, and surveillance systems.Source: Discussion · p. 13

Methods

LE-EpiGNODE represents multiscale epidemic dynamics as a structured UDE with distinct graph supports for age-dependent infection generation and mobility-driven transport. The pathways share epidemic states but retain different state actions and process identities.

Methods

  • The structured UDE combines a mechanism-defined governing vector field with learnable Lagrangian infection-generation and Eulerian population-transport operators.The operators learn process-specific interaction strengths while the governing mechanisms constrain their admissible state transformations.Source: LE-EpiGNODE dynamical formulation · pp. 14–15
  • The two pathways act differently: contact-mediated infection changes compartmental state, while transport redistributes existing states across regions without changing compartment identity.The formulation retains transition direction, dependence on infectious source states, and population conservation during spatial transport.Source: LE-EpiGNODE dynamical formulation · p. 14
  • Each node represents one region–age combination, yielding a shared age–space node set for both graph views.There are M regions, A age groups, and MA total nodes.Source: LE-EpiGNODE dynamical formulation · p. 15
  • The Lagrangian graph connects age groups within a region to represent infectious-source contributions to infection generation in target age groups.The Eulerian graph connects regions for the same age group to represent population movement between regions.Source: LE-EpiGNODE dynamical formulation · p. 15

Process-specific graph neural operators in the UDE formulation

Separate state- and time-dependent graph operators represent infection generation and population transport, with mechanism-defined state actions coupling both pathways through the epidemic state. The construction extends from SIR to SEIR while retaining population conservation.

Process-specific graph neural operators in the UDE formulation

  • Separate graph operators learn state- and time-dependent interactions on the contact and mobility graph supports, coupled through the shared epidemic state.The Lagrangian pathway models infection-generating transitions; the Eulerian pathway redistributes the full compartmental state with zero-sum flux.Source: Process-specific graph neural… · p. 15

Lagrangian infection-generation operator.

  • The Lagrangian operator uses infectious-source and susceptible-target states, age-contact structure, and time to model within-region infection generation.The contact structure is a prior rather than a fixed transmission coefficient, and the effective interaction can vary with state and time.Source: Lagrangian infection-generation operator · pp. 15–16

Eulerian population-transport operator.

  • The Eulerian operator represents spatial connectivity using regional populations and separation, with epidemic activity and time allowing transport intensity to vary.Its zero-sum constraint redistributes existing epidemic states without creating or removing population mass.Source: Eulerian population-transport operator · pp. 16–17
  • The Eulerian pathway distinguishes transport of an existing infectious population from Lagrangian messages representing newly generated infections.Predicted OD mobility is converted into effective per-capita transport rates, with diagonal entries set by outgoing interactions.Source: Eulerian population-transport operator · p. 17

Coupled epidemic dynamics.

  • Fixed transition vectors assign learned interaction strengths to compartment transitions, while the operators remain coupled through the shared epidemic state.The complete vector field combines the two operators with a self-loop progression message.Source: Coupled epidemic dynamics · p. 17

Coupled epidemic dynamics

  • The complete vector field separates infection generation, local compartmental progression, and conservative spatial transport.Learned operators set edge-wise interaction strengths; transition vectors and zero-sum transport structure determine their state actions.Source: Coupled epidemic dynamics · p. 17

Coupled epidemic dynamics.

  • Population conservation is encoded directly in the vector field through zero-sum compartment transitions and Eulerian transport, not an added penalty.Neither mechanism changes the system's total represented population.Source: Coupled epidemic dynamics · p. 18

SEIR extension for real-world epidemics.

  • For real-world COVID-19 analyses, the same graph pathways extend from an SIR backbone to SEIR by adding an exposed compartment.The Lagrangian messages generate susceptible-to-exposed transitions, progression covers exposed-to-infectious and infectious-to-removed transitions, and Eulerian messages transport the full SEIR state; conservation is retained.Source: SEIR extension for real-world epidemics · p. 18

Learning structured epidemic dynamics

LE-EpiGNODE learns structured epidemic dynamics from reported cases while using historical OD observations to inform a frozen mobility network. Forecasts integrate the learned dynamics forward without future case or OD observations.

Learning structured epidemic dynamics

  • Reported incidence is treated as the I → R transition flow, while other compartment states and operator-level interactions remain latent.The continuous-time dynamics are learned indirectly through the reporting channel.Source: Learning structured epidemic dynamics · p. 19

OD-informed mobility pretraining.

  • The auxiliary mobility network is pretrained on historical origin–destination observations, then held fixed while the epidemic model is optimized from reported cases.This two-stage procedure transfers mobility information into the Eulerian pathway before coupling it to the epidemic trajectory.Source: Learning structured epidemic dynamics · p. 19, OD-informed mobility pretraining · p. 19
  • During pretraining, reported cases construct epidemic-activity inputs and the mobility network is fitted to observed OD values; during epidemic training, evolving predicted infectious states supply activity inputs.The frozen network’s predicted transport intensities can still vary with epidemic state and time, making it an OD-informed prior rather than a fixed future mobility sequence.Source: OD-informed mobility pretraining · pp. 19–20

Epidemic training objective.

  • The epidemic objective combines reported-case fit with an initial-observation consistency term while keeping pretrained mobility parameters fixed.Compartment-transition directions and population conservation are encoded in the vector field rather than additional penalty terms.Source: Epidemic training objective · p. 20

Optimization and forecasting.

  • Forecasts integrate the learned dynamics from the forecast origin using evolving predicted states, without requiring future reported cases or future OD observations.The Lagrangian operator updates infection interactions from state, contact prior, and time; Eulerian activity inputs are generated internally.Source: Optimization and forecasting · pp. 20–21

Data

The evaluation combines controlled synthetic systems with three COVID-19 datasets representing age-structured, mobility-driven, and coupled age–space dynamics. Synthetic ground truth supports evaluation of both forecasts and dynamical recovery.

Data

  • The study uses synthetic and real-world datasets to test forecasting, structural contributions, dynamical fidelity, and forecasting from partial observations.Synthetic systems have known generating mechanisms and operators; real-world settings include age structure, mobility, or both.Source: Data · p. 21

Synthetic datasets.

  • Fourteen synthetic conditions combine four infection-generation mechanisms, two transport mechanisms, and eight multiscale pairings.The mechanisms span baseline, temporal modulation, infection-dependent feedback, nonlinear state dependence, and controlled risk-aware transport; epidemic and OD observations include Gaussian noise at 5% of each trajectory maximum.Source: Synthetic datasets · p. 21

Real-world COVID-19 datasets.

  • Real-world validation covers Ontario age-structured transmission, Hubei mobility-driven spatial dynamics, and Spain coupled age–space dynamics using daily COVID-19 observations.Latent states and generating operators are unobserved in these datasets, and the evaluation uses the SEIR extension.Source: Real-world COVID-19 datasets · p. 21

Evaluation design

The evaluation tests long-horizon forecasting across synthetic and real-world settings against multiple baseline families, then examines structural and process-aligned ablations and forecasting and dynamical-consistency metrics.

Evaluation design

  • Synthetic experiments use pre-peak training data and 1-, 4-, 12- and 20-week horizons; real-world training periods and forecast horizons differ by dataset.Ontario uses 25 training days and 7/14/35/67-day horizons, Hubei 15 days and 7/14/26-day horizons, and Spain 20 days and 7/14/49/93-day horizons.Source: Evaluation design · p. 22

Baseline models.

  • The comparator set spans mechanistic, general forecasting, epidemic-specific graph and hybrid mechanistic-learning models, with synthetic and real-world comparisons using different subsets.The study describes 21 comparators overall; synthetic tests use mechanistic and general forecasting baselines, while real-world tests additionally include epidemic-specific graph and hybrid models.Source: Baseline models · p. 22

Ablation settings.

  • Five ablations remove or replace the Lagrangian–Eulerian structure, either pathway, the age-contact representation, or OD-supervised mobility pretraining.The variants retain the same observations and evaluation settings as the complete model except for their specified change.Source: Ablation settings · pp. 22–23

Evaluation metrics.

  • Forecasting is assessed with MAE and WMAPE, while synthetic ground truth enables evaluation of instantaneous operators, state fluxes and integrated interaction structure.Operator and flux agreement uses WMAPE and PCC; integrated LIF and EIF agreement uses SRCC, with EIF excluding diagonal conservation terms.Source: Evaluation metrics · p. 23
  • LIF summarizes infection-generation fluxes across regions, whereas EIF summarizes infectious-state transport across age groups; row and column normalization express target and source distributions, respectively.For real-world data, operator and flux ground truth is unavailable, so integrated fluxes are aggregated and normalized for process-level interpretation.Source: Evaluation metrics · p. 24

Data availability

The study uses publicly accessible empirical case and mobility data, alongside synthetic datasets generated from specified mechanisms.

Data availability

  • Ontario, Hubei and Spain COVID-19 case data and Hubei and Spain OD mobility data are publicly accessible from the listed repositories.Synthetic data use the study’s described mechanisms, with generation equations and parameter settings reported in Supplementary Methods section 1.1 and Supplementary Tables 1–3.Source: Data availability · p. 24

Extended Data

The Extended Data materials document forecasting performance, OD mobility errors and synthetic dynamical fidelity, alongside trajectory and mobility-extrapolation plots.

Extended Data

  • Extended Data Table 1 reports MAE and WMAPE (%) for synthetic-scenario forecasting, marking the best and second-best results.Extended Data Fig. 1 shows coupled multiscale trajectories for four Lagrangian mechanisms paired with BRGM or CRGM; its dashed line marks forecast origin and shading marks the out-of-sample period.Source: Table 1 · p. 25, Figure 1 · p. 28
  • Extended Data Table 2 reports MAE and WMAPE (%) for forecasting on the real-world COVID-19 datasets, marking the best and second-best results.The supplied caption does not give individual table values.Source: Table 2 · p. 26
  • Extended Data Table 3 reports MAE and WMAPE (%) for directed OD mobility values in Hubei and Spain, with MAE on the original mobility scales and not directly comparable across datasets.Extended Data Figs. 2 and 3 plot observed and predicted directed OD trajectories; their overall R2 covers fitting and forecasting, while forecasting errors are reported separately in the table.Source: Table 3 · p. 27, Figure 2 · p. 29, Figure 3 · p. 30
  • Extended Data Table 4 evaluates synthetic dynamical fidelity using operator and state-flux WMAPE/PCC and integrated-flux SRCC; lower WMAPE and higher PCC or SRCC indicate stronger agreement with known dynamics.The table caption identifies LIF-SRCC for the Lagrangian branch and EIF-SRCC for the Eulerian branch.Source: Table 4 · p. 27

S1 Supplementary Methods

The supplementary methods specify a controlled synthetic benchmark, its coupled epidemic dynamics and observation protocol, and the input and comparator configurations used in real-world and synthetic evaluations.

S1.1 Synthetic Data Generation

  • The synthetic benchmark contains 14 conditions across isolated Lagrangian infection generation, isolated Eulerian transport and coupled multiscale dynamics, sampled daily over 0–200 days.Within each structural setting, population sizes, initial states, epidemiological parameters and observation-noise rules are held fixed so trajectories primarily differ through the prescribed mechanisms.Source: S1.1 Synthetic Data Generation · p. 36

S1.1.1 Dynamical System

  • The generating system separates Lagrangian infection effects from Eulerian mobility rates, with the latter redistributing the same age group across regions without changing its total population.The three scenarios respectively disable mobility, fix one age group and local contact intensity, or activate both mechanisms across four age groups and four regions.Source: S1.1.1 Dynamical System · pp. 36–37

S1.1.2 Mechanism Definitions

  • The synthetic conditions are designed to test forecasting and process-level fidelity across stationary, time-modulated, state-dependent, nonlinear and coupled age–space dynamics, rather than reproduce a particular outbreak.The Lagrangian mechanisms govern age-contact infection generation, whereas Eulerian mechanisms govern conservative transport across regions.Source: S1.1.2 Mechanism Definitions · pp. 37–38

Lagrangian Infection Mechanisms.

  • Four Lagrangian mechanisms vary infection generation from stable baseline contacts to explicit temporal modulation, infection-responsive adjustment and nonlinear state dependence.BMCM, TDCM, IACM and NMCM instantiate these distinct dynamics.Source: Lagrangian Infection Mechanisms · p. 38

Eulerian Mobility Mechanisms.

S1.1.6 Observation Noise

  • Observation noise is added to infection states and OD mobility observations, but does not affect the simulated state evolution.The initial infections in Eulerian and multiscale settings are concentrated in the first region, enabling examination of mobility-driven redistribution.Source: S1.1.3 Scenario-level Settings · p. 39, S1.1.6 Observation Noise · p. 41

S1.1.7 Synthetic Data Overview

  • The synthetic overview contrasts isolated infection-generation and transport trajectories and shows coupled age–time and city–time patterns across the eight mechanism pairs.The supplementary figure’s coupled patterns distinguish Lagrangian mechanisms across rows and BRGM versus CRGM within each row.Source: S1.1.7 Synthetic Data Overview · p. 41, Figure 1 · p. 42, Figure 2 · p. 43

S1.2 Real-world Data and Mobility Predictor Settings

S2 Supplementary Results

Across synthetic mechanisms, inferred operators reproduce principal temporal patterns and relative edge structure, with stronger agreement for Eulerian transport than Lagrangian infection generation. The supplementary trajectories compare inferred operators with reference dynamics across Lagrangian, Eulerian, and multiscale scenarios.

S2.1.1 Lagrangian Scenario

S2.1.2 Eulerian Scenario

S2.1.3 Multiscale Scenario

  • Multiscale supplements present reference and inferred FL(t) and FE(t) trajectories for coupled mechanisms across regions and age groups.Each multiscale figure contains four grids, corresponding to Regions 1–4 for FL(t) or Age groups 1–4 for FE(t).Source: S2.1 Dynamical Fidelity: Complete… · p. 50

S2.1 Dynamical Fidelity: Complete Operator Trajectories

  • Inferred operators reproduce principal temporal patterns and relative edge structure, with stronger agreement for FE(t) than FL(t).Larger FL(t) discrepancies appear in early and rapidly varying segments; weaker early constraints, latent-state compensation, smooth parameterizations, and multiscale coupling may contribute.Source: S2.1 Dynamical Fidelity: Complete… · p. 50

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