A graph-based neural operator remained bounded in the study's long numerical rollouts, including an out-of-distribution test in which MPNO completed all 91 autoregressive steps of a single Sod shock-tube trajectory without divergence. FNO diverged at step 57 and WNO at step 4 in that same comparison.
The findings come from an arXiv preprint, version 1, dated 26 August 2026. It reports numerical surrogate-model tests on Burgers flow, Darcy flow, concrete penetration and compressible Euler dynamics.
A stability constraint built into the propagator
MPNO represents one-step evolution as a Markov propagation process on a graph. Its physics-coupled edge weights and normalized graph Laplacian form a propagation matrix, written as P = I − αL̃, whose spectral radius—the largest absolute eigenvalue—is constrained to be no greater than one.
The edge weight combines an interface-coupling term, contact area, traction magnitude and an exponential correction from a neural network. The same recipe changes its material-property input between problems: Darcy uses harmonic-mean permeability, while Burgers uses uniform weights.
In an operator-level check across 10 aggregate distributions, power iteration reached a relative estimation error of about 5 × 10−6 after 20 steps. The measured eigenvalues ranged from 0.423 to 1.000, and all 10 cases were positive semidefinite.
That bound applies to the linear propagation operator. Stability of the complete nonlinear model was assessed through rollouts rather than established by the operator bound alone.
Long rollouts stayed bounded, but accuracy varied
On concrete penetration, MPNO's single-step relative L2 error was 0.7304 ± 0.0008 across three initializations, compared with 0.7210 for FNO. MPNO used about 20,000 parameters. The dataset contained 400 independent effective samples, split into 70% for training and 15% each for validation and testing.
Across 29 concrete rollout steps, MPNO's error remained bounded on all six selected seeds for impact velocities of 100, 135 and 165 m/s, reaching about 0.98 at the final step. WNO showed error explosion, while MGN's energy decayed; FNO was stable in the measured comparison.
On Burgers, MPNO's 30-step rollout remained stable without collapse but ended with relative L2 error of 1.49 and energy retention of 0.21. FNO ended at 1.28 and 0.32, while WNO ended at 4.25 and 1.64, respectively.
The edge-weight recipe was also used in Darcy flow, where the test relative L2 error was 0.558 for MPNO, 0.053 for FNO and 0.434 for WNO. Darcy was used as a steady-state transfer test, not an autoregressive-stability test; the comparison covered 20 test permeability fields.
In the ablation study, the no_spec configuration had a three-case relative L2 error of 0.7366, compared with 0.7304 ± 0.0008 for the baseline. The ablations were single runs with SEED=42, so the small difference is reported with caution.
A single-trajectory Euler test
For the out-of-distribution Euler experiment, the model was trained on 112 trajectories of 21 frames each and tested on one official 101-frame Sod shock-tube trajectory. MPNO and MeshGraphNet completed all 91 autoregressive steps without divergence; MPNO's final relative L2 error was 17.96, compared with 22.74 for MeshGraphNet.
The reported implementation took about 2.4 milliseconds for one CPU inference step and 3.7 milliseconds on GPU. A 29-step rollout took about 100 milliseconds, and the authors reported roughly a 105-fold speed-up over LS-DYNA. Such timing comparisons depend on hardware and implementation.
The evidence remains tied to the tested configurations. Concrete long-horizon statistics used six selected seeds, and the Euler generalization result came from one trajectory; the operator-level spectral bound does not by itself establish high long-horizon amplitude fidelity or accuracy.
Taken together, the preprint reports a measured spectral bound for the linear propagator and bounded MPNO behavior in the examples tested. It does not establish that the full nonlinear system will retain both stability and accurate amplitudes across a wider range of transient problems.
Paper data and sources
Original title: A Constitutive Markov Physics-Informed Neural Operator (MPNO) for Autoregressive Stability in Transient Dynamics
Authors: Wenpu Du, Peng Zhou, Yunlong Xia et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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