Simultaneous inference of environmental and interaction forces in collective dynamics

1 Introduction

• Trajectory-based governing-equation learning can improve prediction beyond direct data extrapolation, while arbitrary vector-field inference becomes high-dimensional and vulnerable to omitted mechanisms.Collective systems often have many agents, unknown functional forms, and environmental or self-propulsion terms that candidate-library methods may miss.Source: §1 · p. 2
• Structured variational methods reduce collective-dynamics inference from a high-dimensional vector field to a low-dimensional pairwise interaction kernel based on known system structure.The kernel ϕ(r) encodes distance-dependent attraction or repulsion, while the framework can extend to systems with other known dimensionality reductions.Source: §1 · p. 2
• The paper extends this framework to jointly infer interaction kernels and intra-agent environmental forces, preserving pairwise structure while covering broad collective behaviors.The environmental force f captures intrinsic effects such as self-propulsion and friction, and the resulting model remains lower-dimensional than a general F: R^D → R^D with D = dN.Source: §1 · p. 3
• The contributions include unified semi-parametric and fully nonparametric first- and second-order learning, force-based and trajectory-based evaluation, sample-complexity and noise-robustness studies, scaling analyses, and mechanistic model selection.The semi-parametric formulation accommodates known environmental-force forms while learning ϕ non-parametrically; model selection identifies active forces from trajectory data.Source: §1 · p. 3

1.1 Related Methods

• Prior variational work established nonparametric interaction-kernel learning for homogeneous, heterogeneous, first-order, second-order, and more general agent-based systems, including convergence results and numerical validation.These methods address fixed or increasing agent counts and can use pairwise distances or other pairwise state quantities.Source: §1.1 · pp. 3–4

2 Methods for learning collective systems

• The framework models first-order agent dynamics with an identical environmental force f(x) and distance-based interaction kernel ϕ, while allowing adaptation to more general kernels.The environmental term may represent friction, self-propulsion, external effects, or intrinsic agent dynamics; second-order analogues are provided separately.Source: §2.1 · p. 4, §2 · p. 4

2.1.1 Trajectory data

• The fully non-parametric method uses M independent initial-condition replicates observed at discrete times, with velocities estimated from filtered trajectories and pairwise distances computed from agent states.The observed trajectories, approximated velocities, and pairwise distances provide the data used to estimate f and ϕ.Source: §2.1.1 · pp. 5–6

2.1.2 Estimation algorithm

• Estimators are obtained by minimizing an average trajectory-prediction error over finite-dimensional hypothesis spaces Hf and Hϕ, reducing the problem to a quadratic least-squares system.The coefficient minimizer is characterized by normal equations, so the environmental and interaction estimates can be computed jointly from multiple replicate-time observations.Source: §2.1.2 · pp. 6–8, §2.1.3 · p. 8

2.1.4 Hypothesis space selection

• Localized basis functions define the hypothesis spaces: distance-partitioned indicators, polynomials, or B-splines represent ϕ, while localized functions on state-space hyper-rectangles represent vector-valued f.Partitions are chosen over observed supports to balance resolution and sufficient observations; localization is intended to improve robustness to noise and outliers.Source: §2.1.4 · pp. 9–10

2.1.6 Computational complexity

• The algorithm assembles the normal-equation system in parallel across replicates and times, with overall complexity O(MLN^2 + n^3), where n = nϕ + nf.Pairwise-distance computation typically dominates, and accumulated matrices and vectors can be updated incrementally for online learning.Source: §2.1.6 · pp. 10–11

2.2 Variational methods for learning collective and environmental forces: semi-parametric approach (first-order)

• In the semi-parametric approach, a prescribed environmental-force form f(x; p) is fitted through its parameter vector p, while ϕ remains non-parametric and uses the same trajectory-data framework.Comparisons with the fully non-parametric approach use identical initial-condition samples.Source: §2.2 · p. 11
• The method minimizes E(p, β) over environmental parameters p and kernel coefficients β, allowing known parameter constraints but generally yielding a non-convex optimization problem.When f depends nonlinearly on p, global and local minimizers may be difficult to establish or compute.Source: §2.2 · p. 11
• Because the objective is quadratic in β, variable projection reduces the nonlinear optimization dimension; Gauss–Newton, Levenberg–Marquardt, or trust-region methods can then solve the reduced problem.Multiple initial guesses are recommended when non-convexity is expected.Source: §2.2 · p. 11
• All systems considered are linear in parameters, so the semi-parametric estimators are obtained from normal equations as a linear least-squares problem.This removes the need to address non-convex optimization for the models studied.Source: §2.2 · p. 11

3 Evaluation of estimators

• The evaluation framework assesses feature recovery, governing-equation consistency, trajectory accuracy, sampling variability, agent scalability, training-data dependence, and robustness to observational noise.These metrics jointly evaluate identifiability, predictive capacity, consistency, and practical applicability.Source: §3 · p. 12

3.1 Induced measures on data

• Accuracy is measured under induced state and pairwise-distance distributions, emphasizing regions explored by observed trajectories through weighted L2(ρR) and L2(ρX) errors.In practice, empirical measures ˆρR and ˆρX approximate the unknown distributions; training-data coverage supports their use for estimation-error assessment.Source: §3.1 · pp. 12–13

3.2 Feature recovery metrics

• Feature-recovery errors quantify how accurately variational estimators recover synthetic systems’ interaction kernels and environmental forces, whereas residual errors measure discrepancies between observed and predicted derivatives.Feature recovery provides mechanistic evidence but generally requires known ground-truth forces; residual errors are data-dependent and are minimized by the variational objective.Source: §3.2 · p. 14, §3.3 · p. 14

3.4 Trajectory metrics

• Trajectory metrics separate reconstruction on training intervals from extrapolation on held-out intervals, while relative RMS errors provide an outlier-resistant fidelity measure used for model selection.Aggregated means summarize accuracy across realizations, and standard deviations quantify variability; prediction-window errors assess predictive performance.Source: §3.4 · pp. 15–16

3.5 Metrics for consistency

• Consistency is evaluated through repeated-trial uncertainty, transfer from N-agent training systems to 4N-agent prediction systems, convergence with more samples or temporal measurements, and robustness to controlled noise.The scalability test does not train on the 4N-agent system, while training-data experiments vary M or L with other settings fixed.Source: §3.5.1 · p. 17, §3.5.2 · p. 17, §3.5 · p. 17

3.5.4 Robustness to noise in observation data

• Noise robustness is assessed by perturbing observations and evaluating force-recovery and trajectory metrics, with moving-average and Kalman RTS smoothing offered as practical preprocessing alternatives.The Kalman smoother estimates states simultaneously and avoids numerical differentiation, whereas moving-average smoothing precedes derivative estimation; a comprehensive filter comparison is outside this work’s scope.Source: §3.5.4 · p. 18

Generalized candidate models

• The procedure compares first- and second-order candidate frameworks containing optional environmental, energy-based interaction, and alignment forces, with learned coefficients indicating supported mechanisms.The framework is restricted to the specified inter-agent particle models, although analogous methods could extend to broader symmetric collective systems.Source: §4 · p. 19

Candidate ranking

• Candidates are ranked by data support and parsimony, prioritizing lower dynamical complexity and fewer active vector-field terms when models explain the data comparably.Each candidate receives a complexity rank based on dynamical order and active features, and its performance is evaluated using the section’s trajectory metrics.Source: §4 · p. 19, Table 1 · p. 20

Data partitioning

• Training data fit candidates, validation data selects among them, and held-out test data is reserved for final evaluation rather than model fitting or selection.The complete trajectory set is partitioned into training, validation, and testing subsets for distinct purposes.Source: §4 · pp. 19–20, Table 2 · p. 21

Compatibility filtering

• A candidate is rejected before ranking if it poorly reconstructs training trajectories or derivatives, cannot be integrated, or has regression-matrix condition number above 10^12.Numerical simulations use τrecon = τres = 0.25 for the reconstruction and residual tolerances.Source: §4 · p. 20

Final selection

• Among admissible candidates, those below the validation trajectory-error threshold are selected; ties are resolved by lower complexity and then lower training residual error, while no adequate candidate may be declared.In numerical experiments, εabs = 10−3 and εrel = 0.05.Source: §4 · pp. 20–22

5.1 Implementation details

• The experiments generate independent training, testing, validation, and measure-estimation samples, use trajectory-derived hypothesis spaces, and evaluate feature, residual, trajectory, and consistency metrics.Ten independent training replicates support reported means and standard deviations; validation is used for model-selection results.Source: §5.1 · pp. 22–23

5.2 Semi-parametric versus fully nonparametric variational learning

• Across three systems, both variants recover interaction mechanisms and trajectories, while semi-parametric modeling primarily benefits environmental-force recovery when its functional form is correct.The comparisons assess mechanistic recovery, consistency, and trajectory accuracy for Kuramoto, SPP, and phototaxis models.Source: §5.2 · p. 23, §5.2.1 · p. 24, §5.2.2 · p. 27

5.2.1 Kuramoto model

• Both methods recover the Kuramoto interaction kernel with relative errors on the order of 10^-2, while the semi-parametric model recovers the environmental force essentially to machine precision.Trajectory errors are small and similar for training and unseen testing initial conditions, with no discernible visual difference between methods.Source: §5.2.1 · p. 24, Figure 1 · p. 24, Figure 2 · p. 25

5.2.2 Self-propelled particle (SPP) model

• For the SPP model, both methods recover the interaction kernel with Erel_ϕ ≈ 0.12, while the semi-parametric environmental-force error decreases by nearly two orders of magnitude.Trajectory errors remain approximately 10^-2, preserving long-time collective behavior, but incorrect environmental-force parameterization produces poor force and trajectory estimates.Source: §5.2.2 · pp. 27, 30, Table 7 · p. 28, Figure 4 · p. 29

5.2.3 Phototaxis model

• In phototaxis, both methods recover light direction, force magnitude, and trajectories accurately, while the semi-parametric environmental-force error is approximately one order of magnitude smaller.Reconstruction and prediction errors for both methods remain on the order of 10^-6 and 10^-5, respectively.Source: §5.2.3 · p. 31, Figure 6 · p. 31, Figure 7 · p. 32

5.3 Scalability

• Because exact environmental-force structure is rarely available, subsequent analyses use the fully nonparametric algorithm with one training sample after repeated-trial reliability was established.Estimator variability as a function of training-data amount is examined later with repeated trials.Source: §5.3 · p. 33
• Learned interaction mechanisms generalize from N to 4N agents, preserving qualitative collective behavior and emergent milling while scaling predictions tenfold and five times beyond training.Semi-parametric errors are generally modestly smaller than fully non-parametric errors, with both formulations capturing dominant mechanisms across scales.Source: §5.3 · p. 33, Figure 8 · p. 34, Table 10 · p. 34

5.4 Dependence on training data

• Increasing replicate diversity M and temporal resolution L improves mechanistic recovery, but M is dominant because new initial conditions explore more state space than denser sampling.With M = 1, increasing L does not significantly improve recovery of ϕ or f.Source: §5.4 · p. 36, Figure 10 · p. 37

5.5 Robustness to noise in observation data

• Under relative noise ζ ∈[0, 0.1], feature recovery gradually deteriorates mainly in poorly sampled regions, while trajectories remain accurately predictable outside training intervals even at ζ = 0.1.Well-supported regions remain largely unaffected by perturbations; corresponding behavior is reported across all systems considered in Section 5.2.Source: §5.5 · p. 36, Figure 11 · p. 38

5.6 Model selection

• The model-selection framework distinguishes competing collective-dynamics structures across second- and first-order systems and identifies the active mechanisms governing each observed trajectory.It evaluates pure alignment, alignment with environmental forcing, combined energy-alignment-environmental mechanisms, and pure interaction opinion dynamics.Source: §5.6 · p. 40

5.6.1 Cucker-Smale model

• For Cucker-Smale data, the framework selects the true second-order alignment-only model, estimating ∥ˆϕA∥L2(ˆρR) =7.422 × 10−1±3.468 × 10−3 versus inactive terms near 10−4.Candidates F2 −S1 fail compatibility, while candidate S2 is selected by minimum complexity among admissible candidates with equal minimum validation error.Source: §5.6.1 · pp. 40–41, Table 12 · p. 41, Figure 13 · p. 41

5.6.2 Phototaxis model

• For phototaxis, model selection identifies alignment ϕA and environmental force f as active while suppressing ϕE, selecting the true alignment-plus-environmental-force framework.The active mechanisms have weighted L2 norms O(10−1), whereas the inactive energy kernel has norm O(10−4).Source: §5.6.2 · p. 42, Table 14 · p. 42, Figure 14 · p. 43

5.6.3 Self-propelled particle model with velocity alignment (SPPCS)

• For SPPCS data, all three mechanisms are recovered with weighted L2 norms O(10−1), and candidate S6 uniquely selects the true energy-alignment-environmental-force framework.Candidates S2−S1 fail compatibility, S2 and S5 are numerically unstable, and S6 achieves the minimum validation trajectory error among admissible candidates.Source: §5.6.3 · p. 44, Table 16 · p. 44, Figure 15 · p. 45

5.6.4 Opinion Dynamics

• For opinion dynamics, the framework identifies a non-zero energy interaction and near-zero environmental force, selecting the first-order pure-interaction model.The algorithm consistently suppresses the inactive environmental term, while all second-order candidates fail the compatibility gate.Source: §5.6.4 · p. 45, Table 17 · p. 45
• The inferred model correctly identifies interaction-dominated first-order dynamics, with interaction-kernel norm 3.586 × 10^-1 ± 1.752 × 10^-2 and environmental-force norm 0.0000 ± 0.0000.Estimation used Tr = 10 independent learning trials.Source: Figure 16 · p. 46
• The procedure selects candidate F1, the first-order energy-only framework, because it has the lowest complexity among candidates with the same minimum validation trajectory error.This model exactly generated the trajectory data, so the observed dynamics are best explained by first-order interactions despite the broader candidate library.Source: §5.6.4 · pp. 45–46, Table 18 · p. 46

Contributions

• The framework extends variational learning by estimating low-dimensional interaction kernels and environmental or intra-agent force laws instead of arbitrary high-dimensional vector fields.This structural formulation yields mechanistic estimators that are computationally tractable and interpretable.Source: §6 · p. 46
• Semi-parametric learning estimates parameters when the environmental-force form is known, whereas fully non-parametric learning recovers both environmental forces and interaction kernels without that prior knowledge.The two formulations provide complementary options based on whether the environmental structure is specified in advance.Source: §6 · p. 47

Results

• The proposed methods accurately recover governing mechanisms and predict collective behavior beyond the training horizon across synchronization, attraction/repulsion, alignment, and externally driven benchmarks.Performance is supported by low feature-recovery and residual errors, accurate trajectory prediction, trial stability, favorable training-data dependence, and robustness to observational noise.Source: §6 · p. 47

Limitations

• Feature recovery is most accurate in well-sampled state-space and pairwise-distance regions, while sparse trajectory coverage limits inference and semi-parametric misspecification can bias estimates.Accurate assumed functional forms can substantially improve parameter recovery, but incorrect structure may produce misleading results.Source: §6 · p. 47

Model selection

• The model-selection procedure identifies active mechanistic components, including environmental, energy-based, alignment, and combined interaction structures, enabling discovery when the dynamics are unknown a priori.Thus, the framework supports mechanistic model discovery in addition to force estimation.Source: §6 · p. 47

Scope and future work

• The approach can extend to structured dynamical systems whose symmetries, invariances, conservation laws, or mechanistic features reduce high-dimensional learning to lower-dimensional functions or feature maps.Future work targets heterogeneous agents, higher-dimensional dependencies, stochastic dynamics, complex environmental couplings, improved computational tuning, and theory of identifiability and recoverability.Source: §6 · p. 47

Fully non-parametric approach (second-order)

• The method infers both environmental and interaction forces from observed trajectories in second-order ODE models of collective dynamics.It considers systems of N particles with agent states x_i(t) and velocities v_i = ˙x_i(t).Source: A Variational methods for learning… · p. 53
• The environmental force f is assumed identical for all agents and, for the systems studied, depends only on velocity rather than position.Thus, f(x_i, v_i) is simplified to f(v_i), reducing its domain from R^2d to R^d.Source: A Variational methods for learning… · p. 53

A.1 Trajectory data

• The method uses full state and velocity trajectories to construct estimators for the interaction kernel ϕ and environmental force f.For second-order dynamics, the state is represented by x and v, with componentwise agent states and velocities.Source: §A.1 · p. 53, §A.2 · p. 54
• Observations comprise M replicate trajectories sampled at discrete times 0 = t1 < t2 < · · · < tL = T.The observed data record each agent’s state and velocity at every sampled time.Source: §A.1 · p. 54
• The estimation algorithm additionally requires accelerations, with ¨x = ˙v approximated from the observed velocity data.These acceleration observations provide the second-order information needed by the learning procedure.Source: §A.1 · p. 54

A.2 Estimation algorithm

• The method minimizes an error functional over environmental and interaction-force hypothesis spaces, reducing the problem to quadratic optimization over coefficient vectors (α, β).Because the objective is quadratic, a solution exists and is characterized through normal equations.Source: §A.2 · pp. 54–55
• For each basis function of f, the second-order estimator uses both x and v and defines the corresponding force representation over agent components.The state and velocity are organized as x = (x1, …, xN) and v = (v1, …, vN), with xi, vi ∈ R^d.Source: §A.2 · p. 54

A.3 Algorithm pseudocode

• Algorithm 3 learns ϕ and f from second-order trajectory data by constructing pairwise quantities and estimating their observed supports.It generates interaction distances, finds minimum and maximum interaction radii, and determines ranges for state and velocity variables.Source: §A.3 · p. 55
• The implementation constructs localized basis functions, assembles the linear system Âθ = b⃗ in parallel over sampled indices, and solves for θ̂.The pseudocode identifies localized bases as the main representation step before system assembly and coefficient recovery.Source: §A.3 · p. 55

B Hypothesis spaces and empirical measures

• The appendix details the hypothesis spaces and trajectory-induced empirical measures used for the model systems considered in Section 5.2.These quantities support the estimation procedures described in Sections 2.1.4 and 3.1.Source: B Hypothesis spaces and empirical… · p. 55
• Across the three systems, the visualized basis functions and trajectory-induced measures instantiate the hypothesis spaces and empirical measures required for learning.The appendix connects these constructions to the model systems analyzed in Section 5.2.Source: B Hypothesis spaces and empirical… · p. 55, §B.1 · p. 56, §B.2 · p. 56

B.1 Kuramoto system

• For the Kuramoto model, Figure 17 visualizes the basis functions used to construct the interaction-kernel and environmental-force hypothesis spaces.The interaction-kernel basis uses linear B-splines with n_ϕ = 10.Source: Figure 17 · p. 56
• The Kuramoto system’s basis functions and induced empirical measures define the quantities employed by the estimation algorithms.These are presented in Figures 17 and 18 as described in Sections 2.1.4 and 3.1.Source: §B.1 · p. 56

B.2 Phototaxis system

• For the phototaxis model, Figures 19 and 20 illustrate the basis functions and induced measures used by the learning algorithm.The visualizations follow the constructions described in Sections 2.1.4 and 3.1.Source: §B.2 · p. 56

B.3 Self-propelled particle (SPP) system

• For the self-propelled particle model, Figures 21 and 22 illustrate the basis functions and induced measures used by the learning algorithm.These quantities are defined according to Sections 2.1.4 and 3.1.Source: §B.3 · p. 56

Metric evaluations

• The evaluations quantify estimation accuracy and examine how measurement noise affects reconstruction.The supplied passage states that tables provide these accuracy and noise analyses, without reporting their numerical results here.Source: C Metric evaluations · p. 56

Kuramoto model

• Kuramoto empirical measures closely agree with reference probability measures for both phase-difference and phase variables, indicating accurate training-data estimates in explored regions.The comparison covers ˆρ∆θ versus ρ∆θ and ˆρθ versus ρθ.Source: Figure 18 · p. 57

Phototaxis model

• Phototaxis empirical measures closely agree with reference measures for position and velocity variables, indicating accurate training-data estimates in explored regions.The comparison covers ˆρR versus ρR and ˆρV versus ρV.Source: Figure 20 · p. 57

SPP model

• SPP empirical measures closely agree with reference measures for position and velocity variables, indicating accurate training-data estimates in explored regions.The comparison covers ˆρR versus ρR and ˆρV versus ρV.Source: Figure 22 · p. 58

Kuramoto model

• For the Kuramoto model at M=256, interaction-kernel errors are 1.228 × 10^-2 for L=801 and 1.224 × 10^-2 for L=51, while environmental-force errors are 6.180 × 10^-3 and 7.115 × 10^-3, respectively.Each entry reports the mean ± sample standard deviation over Tr = 10 independent learning trials.Source: Table 19 · p. 59

Phototaxis model

• The phototaxis model reports interaction-kernel errors around 2.27–2.57 × 10^-4 and environmental-force errors around 1.85–3.66 × 10^-5 across the listed M and L settings.For M=64, environmental-force errors range from 3.392 × 10^-5 ± 3.647 × 10^-6 to 3.657 × 10^-5 ± 5.456 × 10^-6; for M=256, they range from 1.847 × 10^-5 ± 1.588 × 10^-6 to 2.009 × 10^-5 ± 1.905 × 10^-6.Source: Table 20 · p. 60, §C.1 · p. 60

SPP model

• The SPP model is likewise evaluated for interaction-kernel and environmental-force errors across different numbers of replicates M and time samples L, with means and sample standard deviations computed over 10 trials.The supplied passage identifies the evaluation scope but does not provide the SPP table’s cell values.Source: Table 21

C.2 Noise robustness

• Figure 23 visualizes how different observational noise levels affect recovery of mechanisms.Source: §C.2 · p. 61
• Figure 24 shows the corresponding impact of observational noise on trajectory reconstruction and prediction.Source: §C.2 · p. 61

D.1 Effect of initial conditions on inferrence

• Broader initial-condition distributions produce more informative observations and more accurate recovery of the environmental force.This effect is illustrated in Figure 25 by comparing datasets generated from different initial-condition distributions.Source: §D.1 · p. 61
• Narrowly concentrated initial conditions leave portions of the state space poorly sampled, reducing learning accuracy.Source: §D.1 · p. 61

D.2 Basis functions

• The basis-dimension choice is therefore important for balancing approximation accuracy and numerical stability in the nonparametric learning framework.Source: §D.2 · p. 61
• Adaptive basis construction and automatic hyperparameter selection could further improve learning accuracy and computational efficiency.Source: §D.2 · p. 61
• Approximately 10 basis functions minimize the average recovery error for both the SPP interaction kernel and environmental force.Figure 26 varies each basis dimension while averaging over tested values of the other; the lowest average errors occur at approximately ten basis functions.Source: Figure 26 · p. 65, §D.2 · p. 65
• Increasing the basis dimension initially improves approximation quality and reduces recovery error until performance saturates.Source: Figure 26 · p. 65

E.1 Data generation parameters

• Training data estimate interaction kernels and environmental forces, while testing data validate trajectories.Source: Table 22 · p. 65
• The Mρ dataset constructs reference probability densities for the numerical experiments.Source: Table 22 · p. 65
• Table 22 specifies the data-generation parameters used throughout the numerical experiments.Source: Table 22 · p. 65

E.2 Learning algorithm parameters

• Interaction kernels are approximated with radial B-spline bases in the variational least-squares framework.Source: Table 23 · p. 66
• Environmental forces are represented using a tensor product of B-splines.Source: Table 23 · p. 66
• For first-order systems, the environment variable is the state, whereas for second-order systems it is generally the velocity.Source: Table 23 · p. 66

E.3 System initial conditions

• Initial conditions are system-specific: first-order Kuramoto uses u_0 = θ_0, whereas second-order systems use u_0 = (x_0, v_0).Table 24 summarizes the numerical-experiment initial conditions.Source: Table 24 · p. 66
• Phototaxis velocities are initialized from a randomly assigned tensor grid on [−1, 1]^2 with independent Gaussian perturbations.The construction approximately uniformly covers velocity space and improves conditioning of the tensor-product basis for learning f.Source: §E.3 · p. 66
• For SPP and SPPCS, positions are sampled from mixtures of uniformly populated spatial clumps.This defines the cluster-based spatial measure μ_SPP^x.Source: §E.3 · p. 66
• SPP and SPPCS velocities use a perturbed tensor grid on [−2v_eq, 2v_eq]^2, where v_eq = α/β.The resulting construction broadly covers the velocity domain used to learn the environmental force f.Source: §E.3 · p. 66

E.4 Estimates from semi-parametric approach

• Semi-parametric coefficient recovery reports means and standard deviations over T_r = 10 independent trials for each candidate-basis coefficient.Coefficients for mechanisms absent from the model are expected to remain close to zero.Source: Table 25 · p. 67

• The coefficient and metric notation separates fitted environmental and interaction terms while defining residual, feature-recovery, and normalized error measures.θ = (α, β) combines coefficients for the fitted force and kernel; residual errors are normalized by Sres, and feature recovery uses weighted L2 norms.Source: §E.6 · pp. 69, 72
• The hypothesis-space notation defines the environmental-force basis and a data-dependent interaction-kernel domain partitioned by observed pairwise distances.The interaction domain is [Rmin, Rmax], divided into P sub-intervals with nϕ = P × nP localized basis functions, while Hf is finite-dimensional.Source: §E.6 · pp. 69–70
• The notation distinguishes first- and second-order systems through the stacked state-space dimension D and the force input Z.D = dN for first-order systems and D = 2dN for second-order systems; Z = X or (X, V), respectively.Source: §E.6 · p. 71
• The evaluation notation distinguishes training and prediction horizons and records reconstruction, extrapolation, full-trajectory, and replicate-level trajectory errors.[0, T] is the training horizon, (T, Tf] the prediction horizon, and trajectory metrics include sample means, standard deviations, RMS errors, and relative RMS errors.Source: §E.6 · pp. 72–73
• The model-selection notation represents candidate frameworks, compatibility and validation gates, trajectory splits, thresholds, and the selected highest-ranked framework.Candidates pass reconstruction and residual thresholds before validation filtering by τtraj; c∗ denotes the selected candidate.Source: §E.6 · pp. 73–74