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Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies

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Pith's one-line read Two reset strategies — restarting from data or exploiting a limit cycle's contraction — turn one-period accuracy into long-horizon guarantees with error linear in elapsed periods, not double-exponential in the horizon.

desk verdict Honest conditional analysis of two reset strategies; the MPC bound is solid, the Floquet linear-in-period theorem is coherent but its key tube-closeness hypothesis is never met by any deployed model. read the letter →

arxiv 2608.10738 v1 pith:EZ5LI64R submitted 2026-08-11 cs.LG cs.NAmath.NA

Ziqian Li , Nikolaos M. Matzakos This is my paper · ORCID
classification cs.LGcs.NAmath.NA MSC 68T0793B4534C2565L0541A30
keywords NeuralODEslong-timeapproximationmodelpredictivecontrolFloquettheorylimitcyclesSA-NODEPoincaréreturnmaptimehorizonbarrier
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper takes on a specific failure of learned dynamics: the certified error bound of a single neural ODE trained over a long horizon grows double exponentially in the horizon length, which would make long-time prediction look hopeless. It argues this barrier is an artifact of training one monolithic network, not an intrinsic limit of the model class, and develops two training strategies that circumvent it, each built on a 'reset' of the state. The model predictive strategy partitions the horizon into short windows, restarts each window from observed data, and proves the composite model holds a user-chosen tolerance uniformly in time with a total network width that is linear in the horizon. The Floquet strategy targets autonomous systems with a stable limit cycle, trains a single periodic network whose first-return map contracts, and proves that one-period accuracy on a tube around the cycle yields trajectory error bounded by $C\varepsilon(1 + t/\hat T)$ — linear in elapsed periods, with no data needed at deployment. What makes the guarantees usable is that each rests on hypotheses that can be measured on the trained model; the paper measures them, and its own experiments show where the deployed architecture satisfies them and where it does not.

What carries the argument

The argument is carried by two reset mechanisms and the quantities that certify them. The data reset is a model predictive partition: one SA-NODE per window with local time, warm-started parameter copies, and the true state injected at each switch time, which confines the Grönwall factor $e^{L\tau_{\max}}$ to a single window; the load-bearing certificate is the uniform window constant $C_{\tau_{\max},K_\infty,f}$ of Assumption 3.3, finite whenever the reachable tube is bounded with uniformly regular data. The dynamical reset is the Poincaré first-return map $P_\Theta$ of the learned flow: a certified contraction $\rho(DP_\Theta(p)) \le \rho_* < 1$ contracts transverse errors geometrically at every return, while the one quantity contraction cannot control — the period mismatch between the learned and true clocks — accumulates linearly and produces the $1 + t/\hat T$ factor. The Floquet loss trains the surrogate $\tilde\rho_T(\Theta) = \exp\big(\int_0^{\hat T} \operatorname{div} f_\Theta(\hat\gamma_\Theta(t))\,dt\big)$, which by the Liouville–Abel identity is exactly the scalar return-map multiplier in the autonomous plane, so a vanishing Floquet loss implies contraction and hence the linear bound. For the deployed periodic encoding the same surrogate degenerates to the determinant of the monodromy, so the paper computes the full stroboscopic spectrum instead and proves the orbital guarantee of Theorem 4.22, whose two hypotheses are exactly the quantities its training loop already returns.

What would settle it

Train any sequence of SA-NODEs on an autonomous limit-cycle target and measure, for each, the one-period flow error $\varepsilon$ over a tube of radius $r$ around the cycle and the spectral radius $\rho$ of the learned first-return map at its fixed point. A single model with $\rho \le 0.5$, tube-level $\varepsilon \le 0.01$, and stroboscopic error after 100 periods above $10\varepsilon$ would refute the linear envelope $C\varepsilon(1 + t/\hat T)$ of Theorem 4.13; conversely, Proposition 4.20 predicts that for an exactly periodic learned field, stroboscopic contraction $q$ forces one-period error on the cycle to be at least $(1-q)\operatorname{diam}(\Gamma)/2$, so a systematic sweep over trained periodic models would show whether that obstruction is tight.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that long-horizon approximation error for semi-autonomous neural ODEs is governed by the reset mechanism, not by network size. For the data-restarted composite, if training meets a prescribed tolerance $\varepsilon$ on every window and the target's reachable tube is bounded with uniformly regular data, then $\sup_{t,x_0}\|\Phi(t;x_0) - \hat\Phi(t;x_0)\| \le \varepsilon$ with a total width $\lesssim N C^2_{\tau_{\max},K_\infty,f}\,\varepsilon^{-2}$, which is linear in the horizon $T$ for a uniform partition (Theorem 3.5). For an autonomous target with a hyperbolic stable limit cycle, if the learned first-return map is certified to contract, $\rho(DP_\Theta(p)) \le \rho_* < 1$, and one-period flow closeness $\varepsilon$ holds on a tube around the cycle, then every trajectory starting near the cycle satisfies $\|\Phi_\Theta(t;x_0) - \Phi_f(t;x_0)\| \le C\,\varepsilon\,(1 + t/\hat T)$ for all $t \ge 0$: transverse errors are contracted at every return, and only the phase drift between the two clocks accumulates (Theorem 4.13). The paper also proves an obstruction for its deployed time-periodic architecture — an exactly periodic learned field cannot simultaneously have small one-period error and a contracting stroboscopic map — so the deployed models are covered instead by a uniform-in-time orbital bound whose two hypotheses (monodromy spectral radius and orbit distance to the cycle) are directly measurable (Theorem 4.22).

Load-bearing premise

The load-bearing premise is that the trained model's one-period flow error stays below a small $\varepsilon$ uniformly over a whole tube of initial conditions around the target cycle, not just on the single orbit used for training — and the paper's own measurements put its deployed models outside that regime, with tube-level errors hundreds of times the orbit-level error.

Editorial extensions

If this is right

  • A monolithic model trained once on the whole horizon cannot be certified beyond a few multiples of $1/L$: both strategies replace the double-exponential certified width of (2.1) with budgets that grow at most linearly in $T$ (data reset) or that keep a single network with error linear in elapsed periods (dynamical reset).
  • In the data-assisted regime the uniform guarantee is carried by the resets: the same trained windows chained from their own predictions obey only the exponential bound $\varepsilon\,(e^{N\bar L\tau_{\max}}-1)/(e^{\bar L\tau_{\max}}-1)$, so deployment on novel initial conditions needs either observed states at the switch times or dynamical stability.
  • The linear envelope of Theorem 4.13 is attained up to constants by a pure period mismatch: two dynamics with identical radial contraction and angular speeds differing by $O(\varepsilon)$ achieve error at least $c\varepsilon k$ at the $k$-th return until the phase wraps.
  • For the deployed exactly-periodic architecture, trajectory-wise accuracy and stroboscopic contraction are quantitatively incompatible, so the available guarantee is orbital — distance to the target cycle bounded by $\varepsilon_\Gamma$ plus a geometrically decaying transient — and launch-phase fidelity is lost by design.
  • For near-autonomous learned fields the linear bound survives with $\varepsilon$ replaced by $\varepsilon + C\eta$, where $\eta$ is the $C^1$ oscillation of the learned field about its time average; the paper's measured oscillations place its deployed models roughly ten orders of magnitude outside that regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Implicit in the paper but not pursued: if the learned period $T_\Theta$ were explicitly fit to $\hat T$ during training, the dominant linear term of (4.6) would nearly vanish and the error envelope would flatten far below $C\varepsilon(1+t/\hat T)$ for many periods; the polar attainment example of Remark 4.14 makes this a directly measurable prediction — stroboscopic error should track $k\,|T_\The
  • The gap the authors report between tube-level and orbit-level one-period error (median factors of 1124 and 328 in their deployed models) points to the next bottleneck: a training objective that penalizes one-period error uniformly over the tube, rather than only on the base orbit, is the natural route to bring deployed models inside the regime of Theorem 4.13.
  • The obstruction for exactly periodic fields plausibly extends to any entraining learned system: a contracting stroboscopic map forgets the initial phase, so any application that needs phase timing — circadian, cardiac, power-grid models — should track the phase observable explicitly and be content with an orbital guarantee for amplitude.
  • Whether the horizon barrier is intrinsic is left open; if a super-linear width lower bound is ever proved, the MPC composite is optimal in $T$ up to constants, and a longer horizon sweep at fixed tolerance than the four points reported would give the first empirical evidence separating linear from mildly super-linear window growth.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies how the certified error of a learned SA-NODE flow grows with the horizon and proposes two state-reset strategies that avoid the double-exponential barrier of the monolithic certificate (Theorem 2.1; constant (2.1)). The model-predictive strategy partitions the horizon and restarts each window from the true state: under per-window tolerance and a bounded, time-uniformly regular reachable tube (Assumptions 3.1–3.4), Theorem 3.5 yields uniform error ≤ ε with total width O(N C^2 ε^{-2}), linear in T for the uniform partition, while predicted-IC deployment obeys only the exponential bound of Proposition 3.7. The Floquet strategy targets autonomous systems with a hyperbolic stable limit cycle: with a certified contraction of the learned return map (Assumption 4.6) and one-period flow closeness on a tube (Assumption 4.4), Theorem 4.13 gives ||Φ_Θ(t;x0) − Φ_f(t;x0)|| ≤ Cε(1 + t/T̂); Corollary 4.18 links the Floquet loss to the planar multiplier; Remark 4.19 shows the surrogate degenerates to det M for the deployed time-periodic encoding; Propositions 4.20–4.21 prove a quantitative incompatibility of one-period closeness with stroboscopic contraction; Theorem 4.22 supplies the orbital bound (4.15) with measured hypotheses. Experiments: forced Duffing and pendulum for MPC; Stuart–Landau and van der Pol for the Floquet ablation, plus a warm-started autonomous arm.

Significance. The conditional theorems appear sound: Proposition 4.20's three-line proof is correct, the adapted-norm contraction machinery of Lemmas 4.9–4.12 is standard and coherently assembled, Corollary 4.18 constructs the learned fixed point without assuming Assumption 4.6 (a genuine non-circular step), and the MPC budget of Theorem 3.5 follows from its stated assumptions. The paper ships code, models, configs, and figure scripts; reports an independent adaptive-integrator recomputation reproducing the reported multipliers to 10^{-3}; and makes falsifiable quantitative predictions (tube-error lower bounds ε ≥ 0.30 and ε ≥ 1.93; transient decay governed by the measured ρ(M); window count linear in T) that its own measurements confirm. Its candor is exemplary: the demanding character of Assumption 4.4, the measured-not-certified status of Theorem 4.22's inputs, and the grid-wise verification of Assumption 3.4 are all disclosed in place. The significance is nonetheless capped by a wide gap between headline and instantiation: the abstract's 'certified contraction … confines the error to linear growth in the number of elapsed periods' is realized by no trained model in the paper.

major comments (3)
  1. [§4.1 (Assumption 4.4); §5.4] Assumption 4.4 is the load-bearing premise of Theorem 4.13 and Corollary 4.18, and no trained model in the paper is shown to satisfy it. The paper states (Section 4.1, paragraph following Assumption 4.4) that the assumption 'asks for one-period flow closeness at every initial state of the tube N_{2r1}', that a small trajectory loss on the base orbit does not establish it, that a small Floquet loss does not establish it, and that the two certification routes 'presuppose it rather than prove it'. Section 5.4 then quantifies the gap: median tube errors are 0.88 (Stuart–Landau) and 4.84 (van der Pol), i.e. 1124× and 328× the orbit Hausdorff distance ε_Γ, and the measured C^1 oscillation η misses Lemma 4.15's admissible regime by about ten orders of magnitude. The warm-started autonomous arm, the only setting in which the autonomous Theorem 4.13 could apply, has tube errors 0.13 and 0.61–0.78 and is labeled 'not a certified instance: the smallness conditions and constants are not evaluated'. The honest conclusion the paper itself draws in Section 5.4 — 'one-period closeness on the tube, and with it Theorem 4.13 and Lemma 4.15, is not observed at any tuning' — should therefore be reflected in the abstract's Floquet claim, which currently presents the linear bound as the strategy's outcome rather than as an idealized conditional statement.
  2. [§4.4 (Propositions 4.20–4.21); abstract] The obstruction results make the gap structural for the deployed architecture, so the framing issue is not merely empirical. Proposition 4.20 shows that an exactly T̂-periodic learned field with q-contraction of the stroboscopic map on a set containing Γ forces one-period error ε ≥ (1/2)(1−q) diam(Γ); the consistency check in Section 5.4 measures grid Jacobian norms 0.052 (Stuart–Landau) and 0.186 (van der Pol), giving lower bounds ε ≥ 0.30 and ε ≥ 1.93, and the measured tube errors (0.88, 4.84) lie above both. Proposition 4.21 extends the incompatibility to the locally measured ρ(M), and Remark 4.19 shows the scalar surrogate controls det M rather than the spectral radius. Taken together these imply that the linear trajectory bound (4.6) cannot hold for the periodic-encoding models of Section 5.4, regardless of tuning, and that the training target ρ̃_T(Θ) ≤ ρ* is not a stability certificate in that class. The abstract's sentence 'a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods' promises exactly the statement that the paper's own theory rules out for its deployed architecture. The Floquet contributions should be led, in the abstract and Section 1.2, by the orbital guarantee (Theorem 4.22, (4.15)) — the one actually instantiated — with Theorem 4.13 presented explicitly as a conditional result for the idealized autonomous case.
  3. [§4.4 (Theorem 4.22); §5.4 (warm-started arm)] Even the instantiated guarantee is delivered at the level of measurement rather than certificate, which sits uneasily with the paper's 'certified' vocabulary. Theorem 4.22's discussion concedes that the implementation returns 'floating-point eigenvalues of a numerically integrated monodromy and a distance between finite samples, without residual or quadrature bounds, so the reported values are measurements of the hypotheses and not certificates of them', and that the radius r_0 'is likewise not evaluated for the initial conditions used in Section 5.4'. In the warm-started arm the fitted stroboscopic growth 0.09–0.16 per period is described as 'consistent with the envelope Cε(1+k)' of Theorem 4.13(iii), but since C is left unevaluated this is a plausibility check rather than a test of the bound. The paper itself names the remedy (validated integration with interval bounds on both the monodromy eigenvalues and the sampled distances); completing it for at least one representative model per benchmark would convert one instance of each guarantee from 'measured' to 'certified', and the abstract and Section 6.1 should be adjusted accordingly even if that upgrade is not undertaken.
minor comments (6)
  1. [§4.1] Five pairwise distinct spectral quantities (ρ_T(f), ρ*, ρ_Θ, ρ̃_T, ρ(M)) are introduced in a short span; the paragraph distinguishing them is helpful, but a one-line table or footnote at first use would substantially reduce the reader's verification burden.
  2. [§5.4] The phrase 'the operator norm of the Jacobian of the stroboscopic map over the 0.2-tube... is at most 0.052' should read 'the sampled maximum is', since the values are grid evaluations and the next sentence correctly notes they are not a certificate; the current wording could mislead a skimming reader.
  3. [Definition 4.17] The surrogate integrates the divergence over the nominal period T̂ of a curve whose actual period is T_Θ; the proof of Corollary 4.18 Step 4 handles the resulting O(ε) gap, but the definition itself should state that γ̂_Θ denotes a length-T̂ segment of the relaxed periodic orbit rather than 'the curve... sampling one nominal period'.
  4. [§5.2] The first window of the Duffing run has length 10.4, exceeding the training horizon H = 10; the text explains that the stopping rule (3.4) is evaluated on the full remaining interval while training covers [τ_k, min(τ_k + H, T)], but a parenthetical at first mention in the experiment would prevent a misreading.
  5. [§3.1] The phrase 'it makes the warm start Θ_k ← Θ_{k-1} shape-preserving' is used without definition; one clause explaining that local time keeps the time-input bias in [0, τ_max] so that copied weights remain on comparable scales would remove the ambiguity.
  6. [§5.3 (Figure 5.2(a))] The monolithic error is reported to grow as e^{0.39t} with correlation r = 0.88 'over the growth phase', but the growth phase is not precisely delimited; a definition of the fitting window would make the reported rate reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central bounds are conditional on separately stated hypotheses, the apparently tautological MPC statement is explicitly disclosed as conditional, and self-citations supply independent approximation results rather than encoding the conclusions.

full rationale

The derivation chain is not circular in the sense of the patterns above. The MPC uniform bound is presented as conditional on Assumption 3.4, and the paper explicitly says the tolerance 'must be realized on every window and the run must cover the horizon'; the substantive independent content is the summed-width budget (3.8), which follows from the universal-approximation constant of Theorem 2.1 under Assumption 3.3. The Floquet linear bound, Theorem 4.13(iii), has two distinct hypotheses, one-period flow closeness epsilon (Assumption 4.4) and certified return-map contraction (Assumption 4.6), and the conclusion C epsilon (1 + t/T_hat) is neither identical to nor fitted from them; the proof assembles return-map, clock-comparison, and winding lemmas. The paper is unusually transparent about the one place where a hypothesis is not established: Section 4.1 states that 'a small training loss therefore does not establish Assumption 4.4, and neither does a small Floquet loss', and Section 5.4 labels the warm-started autonomous arm 'not a certified instance', with measured tube errors far outside the small-epsilon regime. This is an uninstantiated hypothesis, i.e. a limitation and a correctness risk, not a circular reduction. The cited [16] results supply the SA-NODE approximation theorem and the explicit double-exponential constant; they are parameter-free approximation statements with stated assumptions and are not equivalent to this paper's intended linear-in-horizon or orbital conclusions. The only element that superficially resembles self-definition, the data-IC error bound equaling per-window tolerance, is explicitly qualified as a conditional guarantee and is not used to assert an empirical prediction from a fit.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

Every guarantee in the paper is a conditional theorem: the conclusions follow only when hypotheses such as Assumption 3.4 (per-window tolerance realized), Assumption 4.4 (one-period closeness on a tube), and Assumption 4.6 (certified contraction) hold, and the paper repeatedly measures these hypotheses on trained models instead of claiming they hold automatically. No constants are fitted to make a derivation work; the tolerances epsilon, rho*, and the mesh bound tau_max are user-specified inputs that parametrize the statements. The main provenance note is the reliance on the explicit constant (2.1) from the authors' own prior work [16], which feeds both width budgets and the barrier statement. The empirical rates reported in Section 5 (e^(0.39t), N ~ 1.34T - 4.50, width slope -0.75) are descriptive fits, not inputs to any theorem.

free parameters (4)
  • per-window tolerance epsilon = 0.5 (Duffing, pendulum, van der Pol comparison)
    User-specified target tolerance of Algorithm 3.1 and of the composite bound (3.7); not fitted to data. It parametrizes the guarantee rather than being an input the derivation needs.
  • contraction threshold rho* = 0.30 (Stuart-Landau), 0.05 (van der Pol)
    User-chosen training target of the Floquet loss (4.9); the paper states it is not adjusted after the fact. It appears in the certified-contraction hypothesis (4.5) and in the orbital hypothesis (B1).
  • window mesh bound tau_max = realized mesh 3.00 in pendulum sweep
    The analysis treats tau_max as a fixed, horizon-independent quantity; the linear budget statement uses N >= ceil(T/tau_max). In the experiments it is realized rather than chosen a priori.
  • empirical law constants = e^(0.39t), N ~ 1.34T - 4.50, width slope -0.75
    Least-squares fits reported in Section 5 to illustrate the predicted error laws; they are descriptive only and are not inputs to any theorem or to the certified constants.
assumptions (8)
  • standard math Theorem 2.1: universal approximation of SA-NODEs with the explicit double-exponential constant (2.1)
    Borrowed from [16, Thm 2.3 and Rem 2.6], a published SIAM result co-authored by the present first author; the whole barrier analysis and both width budgets rest on this explicit constant, so provenance matters.
  • standard math Gronwall amplification bound (2.2) and reachable-tube growth (2.3) are individually sharp
    Verified on the linear example x_dot = Lx in Section 2.2; both mechanisms drive the double-exponential constant and define the barrier the two strategies act on.
  • domain assumption Assumption 3.3: bounded forward reachable tube with uniformly bounded time-shifted Sobolev data
    Needed for the window constant C_{tau_max,K_inf,f} to be independent of T in (3.8); it fails for the spreading example x_dot = Lx, where the certified budget becomes super-linear in T.
  • domain assumption Assumption 3.4: trained windows meet tolerance epsilon on every window and the partition covers [0,T]
    The central conditional hypothesis of Theorem 3.5; verified only on finite grids with an unquantified continuous-supremum gap, as the paper states after (3.6). The window cap failing means the theorem does not apply at all.
  • domain assumption Assumption 4.1: hyperbolic orbitally asymptotically stable limit cycle with spectral radius below one
    Defines the class of targets for the Floquet strategy; standard Floquet theory then supplies return maps, adapted norms, and the reference multiplier rho_T(f).
  • domain assumption Assumption 4.4: one-period flow closeness epsilon on the whole tube N_2r1
    The paper states this is 'the more demanding of the two' hypotheses and that no training loss establishes it; the deployed periodic models are measured to violate it, with tube errors 1124x and 328x the orbit Hausdorff distance.
  • domain assumption Assumption 4.6: certified transverse contraction rho(DP_Theta(p)) <= rho* < 1
    The key certified hypothesis of Theorem 4.13; cold-started autonomous training does not achieve it (Section 5.4), while warm-started fine-tuning achieves it on some seeds and benchmarks.
  • domain assumption Assumption 4.2: the encoded field is exactly T_hat-periodic in time
    Enforced by the architecture (4.2); the oscillation eta of (4.4) is measured, and the deployed models miss the small-eta regime required by Lemma 4.15 by ten orders of magnitude.

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Pith. "Pith review of Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies." pith.science (2026). https://pith.science/paper/EZ5LI64R

@misc{pith2026260810738,
  author       = {Pith},
  title        = {Pith review of: Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZ5LI64R}},
  note         = {Machine review of arXiv:2608.10738}
}
read the original abstract

We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.

Figures

Figures reproduced from arXiv: 2608.10738 by the authors.

Figure 3.1
Figure 3.1. The MPC–SA-NODE composite: one SA-NODE Θk per band Ik, warm-start copies (curved arrows), and data resets at the switch times (dashed arrows, mode (3.2)); at deployment the dashed arrows are replaced by (3.3). Algorithm 3.1 Adaptive MPC–SA-NODE training Require: sampled trajectory data (t, x) on [0, T], tolerance ε, width P, epochs Nep, min window smin, window cap S, training horizon H, warm-start flag 1: k ← 0; tle… view at source ↗
Figure 4.1
Figure 4.1. The Poincar´e reset mechanism. (a) Certified contraction ρΘ < 1: successive returns of the learned flow to the section Σ converge geometrically to the fixed point, so transverse error cannot accumulate. (b) The non-resettable error: the learned clock runs at period TΘ ̸= Tˆ, so the phase drifts by k [PITH_FULL_IMAGE:figures/full_fig_p016_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Floquet-regularized training. (a) Geometric setting: the base-point orbit is the training target, and the returns P k Θp to Σ illustrate the contraction ρΘ < 1. (b) Optimization loop: Ltraj pins the base-point orbit to the data, LF penalizes ρ˜T (Θ) > ρ∗, and the monodromy spectrum is computed post hoc (Remark 4.19). proved by induction on n: for n = 1 the function z(s) := ΦΘ(Tˆ + s; x) solves ˙z = fΘ(z, Tˆ + s) = f… view at source ↗
Figures from the paper (4 more)
Figure 5.1
Figure 5.1. Figure 5.1: Forced Duffing equation (5.1), T = 15, ε = 0.5. (a) Sample tra￾jectories, true versus composite; the dotted vertical line marks the switch time. (b) Error over time, supx0 ∥e(t)∥ on the initial-condition grid; the sawtooth peaks are left limits at the switch times. e…
Figure 5.2
Figure 5.2. Figure 5.2: Pendulum sweeps. (a) Error over time at T = 10, logarithmic scale; the dashed guide is the least-squares fit e 0.39 t to the neural ODE baseline. (b) Win￾dow count versus horizon at ε = 0.5, with least-squares fit. (c) Window count versus tolerance at T = 10; the dot…
Figure 5.3
Figure 5.3. Figure 5.3: Certification ablation. (a) Per-seed monodromy spectral radius ρ(M), by system, encoding, and loss; dashed lines are the thresholds ρ∗, the dotted line is ρ = 1. (b) Stroboscopic error across seeds on the van der Pol oscillator: the periodic encoding stays bounded, t…
Figure 5.4
Figure 5.4. Figure 5.4: Comparison on the van der Pol oscillator. (a) Continuous error: the MPC sawtooth against the Floquet transient followed by saturation. (b) Strobo￾scopic error at the period marks. (c) Phase portrait: both learned trajectories shadow the limit cycle Γ. Remark 4.19 in …

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.