A Constitutive Markov Physics-Informed Neural Operator (MPNO) for Autoregressive Stability in Transient Dynamics

Wenpu Du; Peng Zhou; Yunlong Xia; Sinuo Xin; Congcong Zhang; Boyang Zhang; Yi Zhang; Wenzheng Xu

Figure 1: Autoregressive rollout of neural operators on concrete-penetration stress-field prediction (29 steps, frames 11–39 of the 40-frame window; schematic, anchored to measured single-step errors and end-of-rollout error bounds). MPNO (solid blue) stays bounded throughout; WNO (red dashed) diverges to infinity on all seeds; MeshGraphNets (gray dotted) fails to learn (error stays near 1); FNO (orange band) remains stable on the clean test seeds but, unlike MPNO, carries no spectral guarantee.

Parameter	Value
Target	$\phi 500\times 200$ mm (cylinder)
Aggregate fraction / size	42% / 5–25 mm
Aggregate / mortar constitutive	JH-2 (granite) / CSCM (C30)
Projectile / velocity	$\phi 45$ mm ogive steel projectile / 100, 135, 165, 200 m/s (100 each)
Grid resolution / time frames	$64\times 64$ / 40 frames
ITZ	Homogenized dispersion ( $\sim$ 50 $\mu$ m)

Table 1: Parameters of the concrete-penetration dataset

Figure 2: Problem geometry and data. (a) LS-DYNA setup: a

\phi

500

\times

200 mm cylindrical concrete target (Fuller-graded aggregates 5–25 mm, 42% volume fraction, JH-2/CSCM) impacted by a

\phi

45 mm ogive-nose steel projectile (0.8 kg, 100–200 m/s). (b) An LS-DYNA

\sigma_{zz}

stress-field snapshot of the impact zone (seed 5, 100 m/s, frame 20; normalized by 30 MPa, range

\pm 8.5

; peak

\sigma_{zz}

5.039). (c) An example aggregate volume-fraction field

V_{f}

.

Figure 3: MPNO method. (a) Hexahedral-mesh to graph-topology mapping (2-D section schematic; the 3-D framework uses 6-face adjacency,

E\approx 3N

). (b) Spectrally-constrained chain: hexahedral mesh

\to

adjacency

W=T\cdot A\cdot|\mathbf{t}|\cdot\exp(\mathrm{MLP})

(nonneg. symmetric, § 3.4 )

\to

Laplacian

L=D-W

(PSD, § 3.2 )

\to

Rayleigh-quotient power iteration

\hat{\lambda}_{\max}

(

O(E)

online estimate, § 3.2.2 )

\to

normalized

\tilde{L}=L/\lambda_{\max}

(spectrum in

[0,1]

)

\to

Markov propagator

P=I-\alpha\tilde{L}

(

\rho(P)=1

, Property 1)

\to

decoder

\to

hard-masked MSE loss. Unlike the one-sided Rayleigh-quotient estimate (§ 3.2.2 ), the spectral constraint

\rho(P)\leq 1

is closed and exact; this closed constraint is the mechanism underlying autoregressive stability. (c) Edge-weight product construction:

T_{ij}

(acoustic impedance, harmonic mean

2Z_{i}Z_{j}/(Z_{i}+Z_{j})

)

\cdot

A_{\mathrm{contact}}

(contact area, fixed

=1

)

\cdot

|\mathbf{t}_{i}|

(traction magnitude)

\cdot

\exp(\mathrm{MLP})

(zero-init correction)

\to w_{ij}\geq 0

(nonneg., symmetric).

Figure 4: Spectral-radius taxonomy for autoregressive stability. A one-step propagator with spectral radius

\rho<1

contracts (

\|\mathbf{e}_{t}\|_{2}\to 0

),

\rho>1

diverges (

\|\mathbf{e}_{t}\|_{2}\to\infty

), and

\rho=1

is the critical boundary (error bounded,

\|\mathbf{e}_{t}\|_{2}\leq\|\mathbf{e}_{0}\|_{2}

). MPNO’s constructed propagator

P=I-\alpha\tilde{L}

places

\rho(P)

exactly at the critical boundary (Property 1), independent of training. Measured full-model Jacobian spectral radii (gray markers, representative operating point seed 55 frame 10, § 4.4 Fig. 8 ):

\rho(J)=0.83

(MPNO) and

\rho(J)=0.90

(no_spec). Re-measured over six clean rollout seeds,

\rho(J)

is operating-point-dependent (MPNO mean

\approx 1.05

vs. no_spec

\approx 1.13

); the ordering MPNO

<

no_spec persists, but a single operating point does not certify

\rho(J)\leq 1

. POD-DMD

\rho\approx 0.88

–

0.97

(shaded band) is emergent without an explicit constraint. FNO (no guarantee) relies on emergent stability; WNO diverges (observed).

Method	Parameters	Relative to MPNO
LS-DYNA	0 (physics-based, no training)	N/A
FNO	$\sim$ 79,000	2D implementation for evaluation
WNO	$\sim$ 11,000	2D implementation for evaluation
MeshGraphNets	$\sim$ 1,000,000	typical configuration
MPNO (hidden=32)	$\sim$ 20,000	the proposed model

Table 2: Trainable parameter scale of each method

Model	Parameters	Energy retention after 30 steps	1st-step Rel-L2	Final-step Rel-L2
MPNO	4,482	Stable (energy@final=0.21)	0.37	1.49
FNO	57,825	0.32	0.22	1.28
WNO	17,121	1.64 (seed-dependent)	0.43	4.25

Table 3: Autoregressive-stability comparison on the Burgers equation

Figure 5: Cross-PDE autoregressive stability. (a) Burgers equation (wave/shock, 30-step autoregression): WNO diverges (0.43

\to

4.25, red dashed) while MPNO (blue) and FNO (orange) remain bounded (0.37

\to

1.49 and 0.22

\to

1.28). (b) The same MPNO architecture is applied across PDEs by replacing only the material-property variables in the edge-weight formula (Burgers: uniform weight; Darcy: permeability; penetration: acoustic impedance

Z_{m}=1.0

/

Z_{a}=1.495

), with autoregressive stability maintained in every scenario. Single-step accuracies are reported in Table 3 (Burgers) and in the text (Darcy).

Figure 6: Spectral validation of the propagation operator on the cropped impact-zone graph (10 random aggregate distributions,

K=20

power-iteration steps). (a) Relative error of the Rayleigh-quotient power-iteration estimate of

\lambda_{\max}

decays with

K

; average underestimation 0.0005% at

K=20

(max 0.003%). (b) Eigenvalue spectrum of

P=I-\alpha\tilde{L}

(

\alpha=0.577

) lies in

[0.423,\,1.000]\subset[0,1]

(10/10 PSD), confirming

\rho(P)\leq 1

by construction.

Model	Parameters	Single-step TEST Rel-L2	29-step rollout
MPNO	$\sim$ 20K	$0.7304\pm 0.0008$ ( $N=3$ )	Stable (6/6 seeds across 100/135/165 m/s; error 0.85 $\to$ 0.98 bounded)
MPNO–F13	$\sim$ 37K	$0.7278\pm 0.0030$ ( $N=3$ )	Stable (6 clean seeds)
WNO	$\sim$ 11K	$0.7422\pm 0.0017$	Divergent (to infinity on all test seeds)
FNO	$\sim$ 79K	$0.7210\pm 0.0006$	Stable (no spectral-radius structural guarantee)
MeshGraphNets	$\sim$ 1.0M	0.9985 (no learning)	Prediction energy decays with rollout
Note: MPNO–F13 is an improved variant of MPNO that recomputes the edge weights with an edge-MLP at every graph-propagation round (§ 4.3.1 ), letting the physics coupling run through the propagation process rather than being frozen once; with about 37K parameters (about half of FNO) it raises the single-step accuracy to 0.7278, narrowing the gap to FNO to $\pm 0.007$ , while remaining stable long-horizon.

Table 4: Single-step accuracy and autoregressive rollout comparison on the penetration data (single step with

N=3

; 29-step rollout = the entire active horizon from data frame 10)

Figure 7: (a) LS-DYNA

\sigma_{zz}

stress fields of the impact zone at frames

t=10,20,30,39

(seed 205, 165 m/s), showing the penetration-induced stress wave propagating from the impact point; amplitude decays with geometric spreading. (b) Peak

|\sigma_{zz}|

in the impact zone over 40 frames: the stress peaks at

t\approx 8

(10.82, normalized by 30 MPa) upon impact, then decays with oscillations due to wave propagation and reflection.

Figure 8: Pseudospectral transient growth of the full-model Jacobian (seed 55, frame 10,

N=224

).

\|J^{t}\|

(largest singular value of the

t

-step Jacobian power) is shown for MPNO (solid blue) and no_spec (dashed red); the horizontal dashed line marks the stability threshold

\|J^{t}\|=1

. MPNO’s transient peaks at

t=1

(1.58, equal to

\sigma_{\max}(J)=1.58

) then collapses below 1 at

t=2

and decays monotonically to 0.263, reconciling

\sigma_{\max}(J)>1

with asymptotic contraction at this operating point (

\rho(J)=0.83\leq 1

). no_spec retains growth above 1 through

t=6

(

\rho(J)=0.90

). Model spectral radii

\rho(J)

are positioned on the

\rho

-axis taxonomy in Fig. 4 .

Figure 9: Controlled counterfactual: inflating the learned

\lambda_{\max}

. The estimated

\lambda_{\max}

is multiplied by

1\times/6\times/8\times/10\times

before re-normalizing the propagation operator (all else unchanged). MPNO (blue) preserves its stability plateau at every amplification factor (0/6 seeds diverge); no_spec (red) loses emergent stability monotonically, diverging in 2/6, 3/6, and 4/6 seeds at

6\times

,

8\times

, and

10\times

. The divergence of no_spec under spectrum inflation, despite identical stability on nominal data, shows that spectral normalization, not incidental training, underlies MPNO’s stability.

Configuration	Removed component	Three-case single-step Rel-L2	All-case single-step Rel-L2	Conclusion
baseline	(full model)	$0.7304\pm 0.0008$	0.7108	Baseline ( $N=3$ )
no_spec	Spectral normalization	0.7366	0.7135	Consistent degradation across conventions
no_phys	Physics-coupled edge weight	0.7291	0.7256	Three-case neutral; benefits high-velocity cases
no_round_mlp	round_mlps cyclic post-processing	0.7314	0.7201	Three-case neutral; useful at high velocity
no_speed	Velocity coupling	–	0.7175	Useful
no_appnp	APPNP residual connection	–	0.7191	Useful

Table 5: Ablation study (single-step TEST Rel-L2; main baseline is the

N=3

mean, ablations are single runs with SEED=42)

Figure 10: Efficiency and stability relative to baselines. Horizontal axis: inference time per 29-step rollout (log scale); vertical axis: stability type (constructed / emergent / diverging). MPNO is the only model combining a constructed stability guarantee with lightweight inference (

\sim

0.1 s on CPU vs. 4–8 h for the LS-DYNA reference, a

\approx

10

{}^{5}\times

speed-up); FNO is stable but without a guarantee; WNO diverges; POD-DMD is emergent. Parameter counts are listed in Table 2 .

Model	#Params	Single-step rel.-L2 ( $\times 1$ )	Long-horizon completed	Diverged	Final-step error
MPNO (lightweight)	19,973	$8.82\pm 3.05$	91/91	No	17.96
FNO	1,187,556	$6.67\pm 2.73$	57/91	Yes	$>50$
WNO	73,956	$6.68\pm 4.29$	4/91	Yes	$>50$
MeshGraphNet	63,780	$11.79\pm 3.49$	91/91	No	22.74

Table 6: Cross-domain out-of-distribution generalization on the PDEBench 2D compressible-Euler Sod shock-tube (single trajectory, 101 frames; all models share one protocol). rel.-L2 = dimensionless relative

L_{2}

ratio

\|\hat{\mathbf{u}}-\mathbf{u}\|_{F}/\|\mathbf{u}\|_{F}

; “diverged” = the error exceeds 50 or NaN before the rollout ends.

A Constitutive Markov Physics-Informed Neural Operator (MPNO) for Autoregressive Stability in Transient Dynamics

Abstract

Figures & tables

PDE Operator Learning

Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution

Operator Learning Using Weak Supervision from Walk-on-Spheres

Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization

Abstract

PDE Operator Learning

Explore similar work

Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution

Operator Learning Using Weak Supervision from Walk-on-Spheres

Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization