Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics

Mauricio Herrera-Marín

stat.MLSep 30, 2026

Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics

Authors: Mauricio Herrera-Marín

Organizations: Facultad de Ingenier´ıa, Universidad del Desarrollo, Santiago, Chile

Abstract

Neural operators learn maps between function spaces, while hereditary network dynamics are described by Volterra resolvents with non-rational Laplace symbols. We introduce a fractional Laplace neural operator (fLNO) that embeds this structure in the learned map. For commuting excitation--Laplacian pairs, one block graph-spectral layer represents the full linear Volterra solution operator exactly. We establish an expressivity frontier for finite rational realizations: they approximate fractional memory geometrically on compact frequency windows, but cannot reproduce the non-integer critical asymptotics generated by a branch point, and on the half-line the best rational rate is root-exponential. The same theory yields trainable parametrizations that enforce a prescribed stability margin by construction, and a graphon-transfer theorem separates genuine operator consistency from parameter sharing. In a common-data benchmark, positive rational operators can match or exceed fLNO accuracy on finite horizons, whereas in controlled near-critical experiments fLNO recovers the branching coordinate more faithfully with far fewer parameters; unconstrained rational fits can cross the stability boundary, while certified parametrizations cannot. A four-parameter spectral law transfers without retraining from graphs of size 48 to 192 with 0.51--0.62% relative error. Applications to Chilean aftershock sequences and to renewal models for Chile and 21 Italian regions illustrate structured inference with explicit uncertainty. The contribution is an operator-learning architecture in which exact memory structure, physical coordinates and stability guarantees coexist with competitive accuracy.

Figures & tables

Figure 1: Rational approximation of the tempered fractional symbol at $(\alpha,\theta)=(0.7,0.4)$ . (A) On the compact window $[0,20]$ , the positive Gauss–Jacobi approximation converges geometrically. (B) On the half-line, the dyadic positive construction exhibits the root-exponential $e^{-c\sqrt{N}}$ scaling predicted by the rational approximation theory.

Figure 2: (P1) Best $N$ -pole Markov rational error on the tempered symbol over $[0,20]$ versus $\sqrt{N}$ (a range on which the compact-window geometric regime is observed), with the exact atom for reference. (P2) Critical heavy-tail intensity: a rational surrogate matching the shaded fit window to $4\times 10^{-3}$ extrapolates to a spurious positive constant; the truth follows $t^{\gamma-1}$ .

Model	Params.	In-window	Tail	Branch error	Train (s)
fLNO	3	$0.140\pm 0.025\%$	$2.20\pm 0.16\%$	$0.0264\pm 0.0011$	25.2
Positive rational	25	$0.076\pm 0.020\%$	$2.55\pm 0.01\%$	$0.0374\pm 0.0009$	19.3
Unconstrained rational	24	$0.244\pm 0.057\%$	$3.92\pm 0.32\%$	$0.0517\pm 0.0080$	19.5
Linear SSM	49	$0.333\pm 0.086\%$	$3.92\pm 1.46\%$	—	17.8
FNO-1D	4241	$3.54\pm 0.88\%$	$489\pm 123\%$	—	17.6
DeepONet	18817	$11.3\pm 1.4\%$	$1170\pm 119\%$	—	8.0

Table 1: Common-data operator benchmark at

\rho=0.90

. Errors are mean

\pm

standard deviation across two seeds. “Tail” is the relative

L^{2}

error on

(T_{\rm train},2T_{\rm train}]

. Branch error is the absolute error of the explicit branching parameter when one exists and the zero-frequency mass proxy for the unconstrained rational model. No intrinsic branching coordinate is reported for the generic SSM, FNO, or DeepONet baselines. Training times are included for scale but are hardware- and implementation-dependent.

Figure 3: Common-data operator benchmark. (A) In-window and doubled-horizon tail relative errors (log scale). (B) Error in the structural branching coordinate for the three models in which that quantity is explicit or defined by zero-frequency mass. (C) Trainable parameter count. (D) Transfer of one learned, size-independent fLNO parametrization across deployment graph sizes. Error bars in panels A–B denote one standard deviation across the two main benchmark seeds.

$\rho$	Model	In-window	Tail	$\widehat{\rho}$	$\|\widehat{\rho}-\rho\|$
0.70	fLNO	0.035%	0.814%	0.708	0.00845
0.70	Positive rational	0.255%	11.4%	0.810	0.10979
0.90	fLNO	0.101%	3.04%	0.879	0.02068
0.90	Positive rational	0.074%	5.11%	0.863	0.03692
0.97	fLNO	0.090%	2.18%	0.953	0.01696
0.97	Positive rational	0.038%	0.610%	0.885	0.08483

Table 2: Near-critical comparison after convergence-oriented training. The rational model may be more accurate over a finite doubled horizon, while the fLNO retains a substantially more faithful explicit branching coordinate near the boundary.

Figure 4: Criticality and stability diagnostics. (A) Finite-horizon tail error for fLNO and the positive rational pole–residue operator across three true branching ratios. (B) Absolute error in the explicit branching coordinate. (C) Rightmost-pole real parts from 20 unconstrained rational fits in the scarce/noisy stress test; diamonds mark fits that cross the stability boundary. (D) Tail-error distribution split by the resulting structural classification. Panels C–D are a stress test, not a statement about the typical behavior of unconstrained training.

Protocol	runs	unstable fits	max $\Re\,\xi_{\star}$	median tail error
Nominal, full data	8	0	$-0.0106$	$5.96\%$
Scarce/noisy, signed random init.	20	5	$+0.0302$	$31.9\%$

Table 3: Unconstrained-rational stability audit. The scarce/noisy protocol is deliberately a stress test and is reported separately from the nominal benchmark.

sequence	$M_{c}$	$N$	$\widehat{\rho}$ [profile]	$\widetilde{\rho}$ [bootstrap]	$\widehat{p}$	$\Delta$ AIC
Tocopilla	3.3	909	$0.781\,[0.644,0.948]$	$0.838\,[0.692,1.071]$	1.22	$+1.2$
Maule	3.8	1645	$0.787\,[0.738,0.839]$	$0.792\,[0.714,0.881]$	0.80	$-15.9$
Iquique	3.1	1835	$0.652\,[0.611,0.695]$	$0.660\,[0.598,0.727]$	2.28	$-1.7$
Illapel	3.7	1418	$0.722\,[0.648,0.803]$	$0.745\,[0.653,0.840]$	1.72	$+2.0$

Table 4: Scalar branching diagnostics from the single-source relocated Chilean cohort. Profile intervals are the primary uncertainty statement. The basic parametric-bootstrap interval (

B=300

) is a finite-sample sensitivity diagnostic, not a calibrated nominal-coverage interval.

\Delta

AIC compares the tempered and untempered kernels; negative values favor tempering.

Figure 5: Single-source Chilean seismic analysis. (A) Scalar branching ratio from profile likelihood and from the 300-replicate parametric-bootstrap sensitivity analysis; the dashed line is the critical value

\rho=1

. (B) Dependence of

\widehat{\rho}

on increasingly conservative magnitude thresholds for the 365-day window. (C) AIC ladder for the ordered along-strike spatial operator: M0 has no inter-zone mixing, M1 has symmetric nearest-neighbor mixing, and M2 permits separate northward and southward strengths. (D) Estimated symmetric M1 coupling with profile intervals.

sequence	$\widehat{\rho}_{\rm M1}$	$\widehat{\varepsilon}$ [profile]	$\Delta\mathrm{AIC}_{10}$	$\Delta\mathrm{AIC}_{21}$
Tocopilla	0.655	$0.137\,[0.001,0.240]$	$-1.8$	$+0.8$
Maule	0.767	$0.515\,[0.420,0.600]$	$-77.1$	$+1.7$
Iquique	0.597	$0.197\,[0.120,0.270]$	$-18.9$	$+1.9$
Illapel	0.695	$0.641\,[0.510,0.780]$	$-62.5$	$+2.0$

Table 5: Ordered along-strike spatial ladder. The

\varepsilon

interval is a one-dimensional profile interval under M1. Negative

\Delta\mathrm{AIC}_{10}

favors symmetric spatial mixing over independent zones; positive

\Delta\mathrm{AIC}_{21}

means that an additional directional north–south asymmetry is not supported.

Figure 6: Ordered along-strike zones for the four relocated Chilean sequences. Colors denote tertiles of the principal rupture-aligned coordinate; stars are the external reviewed mainshock anchors. The zoning is deterministic and is used only after the sequence-specific completeness filter.

Figure 7: Chile, scalar stage. (E1) Windowed $R_{t}$ from the joint fit (dark, $95\%$ conditional profiles) against the fixed-kernel Cori baseline (light), over the incidence backdrop. (E2) Learned effective kernel against the literature generation interval.

model	params	AIC	$\Delta$ AIC	identified structure
M0 diagonal	27	132 831.9	—	—
M1 mean field	28	132 701.2	$-130.7$	$\widehat{\varepsilon}=0.037$ , CI $[0.033,0.040]$
M2 gravity	29	132 685.0	$-16.2$	$\widehat{\ell}=403$ km, CI $[234,894]$

Table 6: Italy, 21 regions: the certified coupling ladder.

Figure 8: Italy, network stage. (N1) – (N2) Semi-profile deviances for the coupling strength and the gravity range. (N3) The certified reproduction matrix,

r(\widehat{A})=1

by construction. (N4) Certified network reproduction number with

95\%

conditional intervals, coupled against uncoupled.

Figure 9: Italy rolling-origin predictive validation. At each origin, M0 and M2 are refitted using all preceding 28-day windows and the next window is scored by the negative-binomial log score under one-step

R

persistence. Positive values favor the gravity-coupled M2 model. M2 improves on M0 in six of eight origins; the dashed line marks the mean gain of

+0.039

nats per region-day.

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Department of Mathematics, Korea University, 145 Anam-ro Seongbuk-gu Seoul, Republic of Korea.

Fourier Neural Operators Partial Differential Equations Fourier

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We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components. We evaluate LFNO on nine benchmarks, including three ODE systems (Duffing, Lorenz, and Pendulum) and six PDE systems (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, and Navier-Stokes). LFNO significantly outperforms existing operators on ODE systems, where transient dynamics dominate, and consistently surpasses LNO while achieving performance competitive with FNO on PDE benchmarks. Furthermore, LFNO offers improved stability and physical interpretability through its component-wise decomposition. These results demonstrate that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales.

Abstract

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