SPLICE: Sparse Physics Library Identification and Compositional Expansion

Lucas Thil; Jesse Read; Guillaume Doquet; Rim Kaddah

cs.LGOct 12, 2026

SPLICE: Sparse Physics Library Identification and Compositional Expansion

Authors: Lucas Thil, Jesse Read, Guillaume Doquet, Rim Kaddah

Organizations: LIX, ´Ecole Polytechnique · IRT SystemX · Safran Tech

Abstract

Neural operator splitting composes a library of neural operators, one per physical mechanism, into zero-shot simulators of new equations. Existing approaches, however, require these mechanisms and their labels to be specified in advance. We ask whether a pretrained library can instead be adapted and grown from unlabeled trajectories in which multiple mechanisms act simultaneously. We introduce SPLICE, a framework that discovers operators without mechanism labels on mixture data. The contributions of the candidate operators to each trajectory are inferred by solving a convex optimization problem. A calibrated test then assigns each discovered operator the mechanism it implements. Failure cases are also detectable, when the library lacks the required physics or when mechanisms are insufficiently separable. Across one and two dimensional systems, the discovered operators recover mechanisms from unlabeled mixtures and compose into zero-shot simulators of unseen equations; a frozen library can also grow an operator for physics absent from pretraining, without labels and without forgetting. On Navier-Stokes, the discovered operators compose about as accurately as their supervised counterparts.

Figures & tables

Figure 1: The SPLICE pipeline. Build trains one operator per labeled effect and composes them, following Serrano et al. (2026) ; Adapt , Grow , and Identify are ours. Adapt learns operators from unlabeled trajectories, and Grow adds an operator when the data reveal dynamics the library cannot express. Identify tests each learned operator against reference solvers used only for evaluation, never for training. At test time, a beam search selects which operators of the library to compose for each unseen equation.

Panel of Figure 2 — Figure 2: The shared architecture and where each stage acts. (a) Every library entry is one learned time-stepper, in which an encoder maps a context window to a code $\theta$ , a hypernetwork turns $\theta$ into the weights of a vector field $f_{\theta}$ , and RK4 advances one step, so one region of code space is one operator. (b) Build trains the library on labeled pure-mechanism trajectories. (c) Adapt and Grow alternate a convex assignment (w-step) with operator refinement ( $\phi$ -step) on unlabeled mixtures, and reseed an operator from the worst-reconstructed window. (d) Compose beam-searches Strang compositions of library operators. (e) Identify grades each discovered operator against reference solvers on a fixed probe set. Blue nodes are data; grey nodes are learned modules.

Architecture	Library	Mean
control architecture	$\mathcal{L}^{\mathrm{SL}}_{2}$	0.190
control architecture	$\mathcal{L}^{\mathrm{SL}}_{8}$	0.096 $\pm$ 0.001
NOS operator	$\mathcal{L}^{\mathrm{SL}}_{8}$	0.096 $\pm$ 0.005
full NOS method	$\mathcal{L}^{\mathrm{SL}}_{8}$	0.107 $\pm$ 0.009
monolithic FNO, no composition (2.2M)	trained on Prim-1D	0.117 $\pm$ 0.001
monolithic FNO, parameter-matched (16.4M)	trained on Prim-1D	0.113 $\pm$ 0.002

Table 1: Zero-shot composition on Eval-9 (mean NRMSE over nine equations; one shared beam harness; all models supervised).

\mathcal{L}^{\mathrm{SL}}_{8}

rows are mean

\pm

sd over 3 seeds; the NOS rows are measured at 3 seeds (Appendix E ) and are indistinguishable from the control architecture within run-to-run variance. The monolithic rows roll out one network trained on the same corpus.

\mathcal{L}^{\mathrm{SL}}_{2}

and

\mathcal{L}^{\mathrm{SL}}_{8}

are supervised libraries trained on two (advection, diffusion) or eight mechanism families. The main observation is that generalizing to new unseen equations is made possible thanks to the library breadth (per-equation results in Table 9 ; architectures defined in § 4 ).

Method	Recovery	Held-out selection	Utility
amortized assignment (best of 4 formulations, Appendix C.1 )	0/5	—	0.377
SPLICE, single run (Eqs. 2 – 3 )	4/5	57.9% $\pm$ 7.7 (51%)	0.147
SPLICE, full (ensemble pooling, predictive $K$ )	5/5	95.0% (46%)	0.154
PDE-FIND (true symbolic terms given) ( Rudy et al., 2017 )	5/5	—	0.056
LEADS-style (environment ids given) ( Yin et al., 2021 )	4/5	—	—

Table 2: Discovery on Mix-1D , five hidden mechanisms. Recovery : mechanisms identified out of five (§ 3.5 ). Held-out selection : accuracy of the active mechanisms predicted on unseen mixtures (permutation chance in parentheses). Utility : composed rollout error on held-out mixtures (lower is better). — marks a quantity the method does not produce. The full method uses the NOS architecture with its codebook; Table 11 (appendix) reports both architectures, both gradings, and oracle utilities.

Training data	Released NOS	Supervised (ours)	Label-free (ours)	Identified
pure mechanisms, $128^{2}$	0.0045	$0.0036\pm 0.0006$	$0.0038\pm 0.0001$	all seeds
pure mechanisms, $256^{2}$	0.0056	$0.0031\pm 0.0004$ ( $0.0033$ )	$0.0028\pm 0.0025$ ( $0.0016$ )	10/10
coupled flows, $256^{2}$	n/a	n/a (no labeled data)	$0.0116\pm 0.0014$ ( $0.0118$ )	10/10

Table 3: Zero-shot Navier–Stokes composition (NRMSE on the released test trajectories, 10 steps, lower is better; persistence scores

0.012

128^{2}

and

0.017

256^{2}

). Supervised : trained on labeled pure-mechanism trajectories. Label-free : discovered from the same trajectories with mechanism labels hidden, or from coupled flows where no pure trajectory exists; labels enter only post-hoc identification. Mean

\pm

sd over seeds, median in parentheses. NOS is used composed, like our libraries; its direct rollout reaches

0.021

128^{2}

and

0.0016

256^{2}

(Appendix E ). Full comparison in Table 14 .

Panel of Figure 3 — Figure 3: Zero-shot composition with a label-free library.Left: exact solution (top), rollout composed from discovered operators (middle) and their difference (bottom), at four times of a 60-step rollout on a coupled flow never seen in training ( $128^{2}$ ). Reading each column from top to bottom, the positions and shapes of the vortices match throughout, and the error grows mainly in intensity. The true flow fades because viscosity and drag dissipate its energy; the composed rollout does not, because the beam selects only advection operators and each operator normalizes the amplitude of its input window, so this dissipation is not represented (Appendix C.3 ). Right: similarity between mechanism actions, for exact (top) and discovered (bottom) operators. Euler is nearly orthogonal to the overlapping smoothing mechanisms, hence identifiable (§ 5.6 ).

Method	Step 1	Step 10	Diffusion	GPU h
Frozen, no growth	$0.221$	$0.88$	$1.00\times$	0
Retrain with replay	$[0.061,0.073]$	$[0.19,0.22]$	$[1.06,1.42]\times$	9
Frozen + new operator (ours)	$[0.057,0.060]$	$[0.32,0.34]$	$1.00\times$	1.5

Table 4: Adding Euler to a network pretrained without it, without labels. Errors are composed relative

L^{2}

on held out Euler windows at rollout steps 1 and 10; diffusion error relative to the original library. Ranges over three seeds. Full comparison in Table 13 .

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$u_{t}\in\mathbb{R}^{C\times N}$ (1D), $\mathbb{R}^{C\times N\times N}$ (2D)	discretized field at step $t$ ; $C$ channels, $N$ grid points/axis
$u_{0:T}$	a trajectory; $u_{t-L:t}$ a length- $L$ context window
$A_{m}$ , $c_{m}\geq 0$	mechanism $m$ (diffusion, advection, …) and its coefficient
$E$	encoder: context window to latent code
$\theta=E(u_{t-L:t})\in\mathbb{R}^{d}$	latent code (selects mechanism and coefficient); $d$ its dimension
$f_{\theta}$	conditioned vector field; continuous dynamics $\mathrm{d}v/\mathrm{d}s=f_{\theta}(v)$

Appendix

Table 5: Symbols (top) and architectures with their instances (bottom).

Name	Content	Role	Labels
Prim-1D	1D, 8 mechanisms $\times$ dense coefficients (spectral solver): advection,	SL training	yes
	diffusion, dispersion, nonlinear advection, decay, hyperdiffusion,
	cubic advection, bistable reaction
Mix-1D	1D, hidden random 1–2-mechanism mixtures of 5 Prim-1D mechanisms	SSL training	hidden (grading only)
	(advection, diffusion, dispersion, nonlinear advection, decay)
Eval-9	1D, nine named PDEs (Table 7 ), each a 2- to 4-term	zero-shot test	—

Appendix

Table 6: Datasets (top) and operator libraries (bottom). “Labels”

=

mechanism identity per trajectory.

Equation	PDE	Active mechanisms
Burgers	$\partial_{t}u=-u\,\partial_{x}u+\nu\,\partial_{xx}u$	nonlinear advection, diffusion
KdV	$\partial_{t}u=-u\,\partial_{x}u-\partial_{xxx}u$	nonlinear advection, dispersion
mKdV	$\partial_{t}u=-u^{2}\,\partial_{x}u-\partial_{xxx}u$	cubic advection, dispersion
Allen–Cahn	$\partial_{t}u=\nu\,\partial_{xx}u+\rho\,u(1-u^{2})$	diffusion, bistable reaction
advection+reaction	$\partial_{t}u=-c\,\partial_{x}u+\rho\,u(1-u)$	advection, reaction
damped KdV	$\partial_{t}u=-u\,\partial_{x}u-\partial_{xxx}u-\gamma u$	nonlinear advection, dispersion, decay

Appendix

Table 7: The nine Eval-9 equations, each a composition of Prim-1D mechanisms. Free coefficients are jittered within the ranges of Table 8 ; the dispersion coefficient is fixed to unity. Signs and the exact reaction terms follow the data generators.

	Setting	Value
1D	grid, steps per trajectory	$n_{x}{=}256$ , $100$ steps ( $L{=}16$ , $T{=}4$ , $\Delta t{=}0.04$ )
	coefficient ranges	advection $[0.1,2.0]$ , diffusion $[0.01,0.4]$ , dispersion $[0.1,1.0]$ ,
		nonlinear advection $[0.1,0.5]$ , decay $[0.1,1.0]$
	training	$10^{5}$ iterations, batch $32$ , lr $3{\times}10^{-4}$ (cosine), weight decay $10^{-4}$
	model	$16.30$ M parameters
2D	grid, snapshots	$128^{2}$ (generated at $256^{2}$ ), $50$ snapshots ( $T{=}4$ , $\Delta t{=}4/250$ )

Appendix

Table 8: Data generation and training configurations. 1D: spectral solver, five discovery mechanisms shown (the full Prim-1D corpus has eight). 2D: pseudo-spectral vorticity solver; data generated at

256^{2}

and trained at

128^{2}

. Parameter counts measured from checkpoints.

Equation	NRMSE	Equation	NRMSE
Burgers	0.091	damped KdV	0.112
KdV	0.123	hyperviscous KdV	0.123
mKdV	0.048	KdV-Burgers-decay (4-term)	0.107
Allen-Cahn	0.074	adv.-diff.-decay-hyperdiff. (4-term)	0.067
advection+reaction	0.115	mean	0.096

Appendix

Table 9: Per-equation composed NRMSE on Eval-9 (NOS operator,

\mathcal{L}^{\mathrm{SL}}_{8}

; mean over 3 seeds, re-measured 2026-09, Appendix E ).

Figure 4: Zero-shot composition of the four-term Eval-9 equation KdV-Burgers with decay (nonlinear advection, dispersion, diffusion, decay) by a five-family supervised library (space-time; the upper-median of 8 test trajectories by composed rollout error, relative $L_{2}$ $0.068$ , set median $0.065$ ). Rows: exact solution; composed rollout; signed error; per-operator increments on their own scales. Compositions are capped at three operators, so one of the four terms must be dropped; the beam keeps dispersion, nonlinear advection, and decay and drops diffusion, a dissipative-for-dissipative substitution consistent with the separability limits of § 5.6 . The nonlinear operator carries the steepening.

Figure 5: Rollouts on three held-out 1D mixtures (§ 5.3 ): ground truth; oracle operators; discovered library $\mathcal{L}^{\mathrm{SSL}}$ ; absolute error (shared scale). Top: advection ${+}$ diffusion, cleanly reproduced ( $0.094$ ). Middle: a traveling advection ${+}$ nonlinear case, where the discovered library reproduces the transport ( $0.076$ ). Bottom: an advection ${+}$ dispersion case at the dispersion-limited boundary ( $0.147$ ): the discovered rollout cannot travel because the residual transport there is dispersion-driven and dispersion is the rarest mechanism in the discovered pool (Appendix C.3 ).

Objective control	rollout	Data control	rollout	one-step
Abl-2D : persistence (floor)	0.93	small self-generated corpus ( Abl-2D )	0.31	0.155
Abl-2D : one-step training	0.965 (diverges)	NOS corpus ( NS-2D )	0.0034	0.0005
Abl-2D : + best post-hoc stabilizer	0.267	NOS model, same protocol	0.021	—
Abl-2D : rollout curriculum	0.309 (stable)	disentanglement (matched budget):
NS-2D : one-step training (pushforward)	0.0225 (stable)	NS-2D , full volume, $128^{2}$	—	0.0031
NS-2D : rollout curriculum (pushforward)	0.038	NS-2D , 256 traj. (volume matched), $128^{2}$	—	0.0007

Appendix

Table 10: Navier–Stokes quality ablations (per-step-mean NRMSE,

128^{2}

). Abl-2D rows: 16-step rollouts, persistence

0.93

. NS-2D data-control rows: released test trajectories over the 10-step released horizon (persistence

0.012

); the NS-2D pushforward rows in the objective control are 16-step rollouts on the same trajectories. Left: training-objective control on Abl-2D and, below the rule, on NS-2D (both objectives from scratch with the pushforward trick). Right: data control and, below the rule, its split into volume, resolution, and solver at a shorter matched budget (hence the

0.0031

anchor in that block). All models supervised.

Method	Architecture	Recovery		Held-out	Utility	Oracle	Labels
		orig.	discr.	selection
amortized (best of 4 formulations, Appendix C.1 )	control	2/5	0/5	—	0.377	0.063	none
exact-fit (Eqs. 2 – 3 ), single run	control	4/5	4/5	57.9% $\pm$ 7.7 (51%)	0.147	0.063	none
+ ensemble pooling + predictive $K$	control	5/5	3/5	72.5% (61%)	0.124	0.063	none
+ ensemble pooling + predictive $K$	NOS + codebook	1/5	5/5	95.0% (46%)	0.154	0.077	none
PDE-FIND (true terms in candidate basis)	—	5/5	5/5	—	0.056	—	symbolic prior

Appendix

Table 11: Complete discovery table on Mix-1D (5 hidden families), expanding Table 2 : recovery under both the original (orig.) and the discriminative grading (Appendix B.2 ), held-out selection accuracy on unseen mixtures (permutation chance in parentheses), and utility through the shared composition harness against each architecture’s oracle. The LEADS rows use a four-environment variant built for LEADS (identities given to LEADS only); — marks cells undefined for a method’s output and n.m. cells not measured, and the amortized row’s selection is omitted as incomparable (fixed top-2 versus variable-cardinality sets).

Figure 6: Discovered operators in operator-signature space (t-SNE) for the five-mechanism Mix-1D library, colored by graded mechanism; stars are the exact hand-coded operators. Discovered operators cluster near their mechanism’s operator; all five mechanisms are recovered, with dispersion the rarest (a single discovered operator).

Dataset	Beam selection	Ours (zero-shot)	Trained baselines
shear_flow (3/4 traj.)	diffusion	0.77	CNextU-Net 0.81; FNO 1.19; TFNO 1.47; U-Net 3.45
shear_flow (1/4 traj.)	Euler	7.5 (1.39 at 10 steps)
rayleigh_benard	none engaged	declines to compose

Appendix

Table 12: Transfer to The Well with no fine-tuning. VRMSE (predicting the mean scores

1.0

), time-averaged, 50-step horizon unless noted; baselines are trained on the dataset, our library never sees it.

Arm	Euler	Step 1	Step 10	Diffusion	GPU h
Frozen, no growth	no	$0.221$	$0.88$	$1.00\times$	0
Retrain, with replay	2/3	$[0.061,0.073]$	$[0.19,0.22]$	$[1.06,1.42]\times$	9
Retrain, no replay	invalid	n/a	n/a	$[60,67]\times$	9.5
New operator, labels	3/3	$[0.053,0.058]$	$[0.39,0.43]$	$1.00\times$	1.5
New operator, label free	3/3	$[0.057,0.060]$	$[0.32,0.34]$	$1.00\times$	1.5
Full retraining	yes	$0.038$	$0.127$	n/a	29

Appendix

Table 13: Adding Euler to a network pretrained without it. Euler : identified seeds (§ 3.5 ). Errors are composed relative

L^{2}

on held out Euler windows at rollout steps 1 and 10; diffusion error is relative to the original library. Ranges over three seeds. New operators have width 16 and are trained on single steps.

supervised (labeled pure-mechanism trajectories)
Model	Composed, $128^{2}$	Composed, $256^{2}$
persistence (repeat last frame)	0.012	0.017
released NOS checkpoint [ Serrano et al., 2026 ]	0.0045	0.0056
NOS operator, this work (codebook, in-context omitted)	0.0036 $\pm$ 0.0006	—
NOS operator, trained at $256^{2}$ (this work)	—	0.0031 $\pm$ 0.0004
full NOS method, retrained (this work)	0.0053	0.0043

Appendix

Table 14: Zero-shot Navier–Stokes on the released NS-2D test trajectories (

n{=}32

; per-step-mean NRMSE over the released 10-step horizon; one protocol for all models). The slice evolves slowly, so every row is read against the persistence floor. “Composed”: operators trained on pure mechanisms only, combined by beam-searched Strang splitting. The

256^{2}

column holds models trained or adapted at native resolution. Rows are mean

\pm

sd where multi-seed; NOS is its single released checkpoint, the retrained full method a single run; the claim is parity. The

256^{2}

-trained NOS-operator row (3 seeds) lands below both references; with single-run references the claim is parity, not superiority. † the full label-free loop of § 3.3 at native

256^{2}

, run at its registered budget: median composed over 10 seeds at the pre-registered label-free-selected round, both mechanisms identified on 10 of 10 seeds at that round; 7 of 10 seeds individually beat their supervised warm start. A secondary rollout-based selector reaches median

0.00093

but identifies both mechanisms on only 7 of 10. A lighter adaptation-only variant reaches

0.0079\pm 0.0003

(Appendix C.3 ).

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Neural operators provide data-driven mappings for modeling dynamical systems. Extending them to families of systems typically requires explicit conditioning variables such as physical parameters, geometries, or boundary conditions. In many real-world settings, these quantities are unobserved. Here, we formulate neural operator discovery (NOD) as the problem of learning both shared solution operators and system-specific variation directly from heterogeneous trajectories without access to labeled governing factors. We introduce a factorized latent-conditioning formulation that jointly learns a neural operator and a low-dimensional latent representation through factorized prediction, trajectory-decoupled sampling, and dimension selection. Across diverse systems, the learned latent representation captures the intrinsic dimensionality of system variation and organizes system instances in a smooth and approximately invertible latent structure aligned with the underlying governing factors. This organization enables generalization to previously unseen system instances, including zero-shot extrapolation across regimes and stable long-horizon prediction. These results establish an interpretable paradigm for operator learning in the absence of explicit factor supervision.

Abstract

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