cs.LGOct 12, 2026

Structure-preserving fidelity diagnostics for partial differential equations

Authors: Kyriakos Flouris

Organizations: MRC Biostatistics Unit, University of Cambridge, Cambridge, United Kingdom · King’s College, University of Cambridge, Cambridge, United Kingdom

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Abstract

Machine learning surrogates of partial differential equations (PDEs) are commonly trained on data generated by numerical solvers. Therefore, the training data obey not the PDE but the modified equation of the numerical scheme, and the test loss, computed on data from the same solver, cannot distinguish a model that learned the physics from one that learned the discretization error. Here, we show that the difference is nevertheless visible in the trained model. We propose a suite of structure-preserving fidelity diagnostics, simple scores that measure whether a learned model respects one known property of the physics, such as a conserved quantity, a symmetry or a Hamiltonian structure, without access to the training data or the governing equation. Over five systems, the diagnostics track the error with respect to the true dynamics with a rank correlation of up to 0.940.94, while the test loss is uncorrelated or anti-correlated with it. For the Burgers, Korteweg-de Vries and Navier-Stokes equations, the diagnostics separate numerical schemes that the test loss ranks as equivalent, including schemes of the same order. order.

Figures & tables

diagnostictests whether the modelrequiresrolloutdetects, spread across learned models
Conservation
Da[Q]D_{a}[Q]changes a quantity QQ at the rate of the physicsQQ and its ratenoartificial viscosity and dispersion, Burgers 9.0×1079.0\times 10^{7} ; non-conservative forms, KdV 123123 ; energy–enstrophy pair, Navier–Stokes 5.65.6 ; Euler schemes, oscillators 4.5×1034.5\times 10^{3}
Ddrift[Q]D_{\mathrm{drift}}[Q]keeps QQ along its own rolloutQQyesnon-symplectic schemes, oscillators 9.1×1039.1\times 10^{3}
Symmetry
Dc[g]D_{c}[g]commutes with a symmetry ggthe symmetry ggnodispersive error, Burgers 6.0×1036.0\times 10^{3} (blind to viscosity); time reversal, oscillators 1.4×1031.4\times 10^{3}
Structure
Table 1: The diagnostic suite. Each diagnostic is computed from the trained model alone, Eq. ( 2 ), and requires only the listed property of the physics. Last column: what the diagnostic detects in this work, with its spread (largest over smallest value) across the learned models of each system. The last two rows are diagnostics that cannot detect a solver (Section III.4 ).
Figure 1: Test loss against fidelity diagnostics. Each marker is one learned model (scheme and seed); both axes are relative to the model trained on the reference scheme ( JAJ_{A} for Navier–Stokes). (a) Test loss and (b) the largest diagnostic of the suite against the transfer error. (c) Spearman rank correlation with the transfer error per system, for the test loss (open) and the diagnostic (filled). Navier–Stokes (grey) is discussed in Section IV.4 .
Figure 2: Linearized learned Burgers models. (a) Effective viscosity at uˉ=0\bar{u}=0 ; (b) relative error of the phase speed at uˉ=0.6\bar{u}=0.6 . Markers: learned models trained on data from each scheme (median over three seeds). Lines: analytic values of the scheme; dashed: the PDE. The upwind line (grey, dotted) is the averaged prediction ν+12⟨∣u∣⟩h\nu+\tfrac{1}{2}\langle|u|\rangle h . Shaded: wavenumbers at which the training data carry less than 10−610^{-6} of the peak power, so the model is not constrained by the data.
schemeDjD_{j}Da[M]D_{a}[M]Da[P]D_{a}[P]Da[H]D_{a}[H]Da[P]D_{a}[P] of the learned model
spectral< ⁣10−30<\!10^{-30}< ⁣10−30<\!10^{-30}< ⁣10−30<\!10^{-30}< ⁣10−30<\!10^{-30}7.7×10−97.7\times 10^{-9}
skew, 4th8.8×10−128.8\times 10^{-12}< ⁣10−30<\!10^{-30}< ⁣10−30<\!10^{-30}7.9×10−137.9\times 10^{-13}6.6×10−96.6\times 10^{-9}
skew, 2nd2.6×10−112.6\times 10^{-11}< ⁣10−30<\!10^{-30}< ⁣10−30<\!10^{-30}2.7×10−82.7\times 10^{-8}1.8×10−91.8\times 10^{-9}
flux, 2nd1.4×10−91.4\times 10^{-9}< ⁣10−30<\!10^{-30}5.6×10−85.6\times 10^{-8}2.0×10−92.0\times 10^{-9}5.6×10−85.6\times 10^{-8}
advective, 2nd7.3×10−97.3\times 10^{-9}< ⁣10−30<\!10^{-30}2.2×10−72.2\times 10^{-7}2.1×10−72.1\times 10^{-7}2.2×10−72.2\times 10^{-7}
Table 2: KdV, N=127N=127 . Diagnostics of the numerical operators for the structure ( DjD_{j} ), mass ( MM ), momentum ( PP ) and energy ( HH ), averaged over 16 probe states. Values below 10−3010^{-30} are round-off. Last column: momentum diagnostic of the model trained on data from each scheme (median over three seeds).
Figure 3: 2D Navier–Stokes, 48×4848\times 48 , ν=0.05\nu=0.05 . (a) Energy and enstrophy diagnostics of the four second-order schemes and the spectral reference; values below 10−2710^{-27} (shaded) are round-off and shown at 10−3010^{-30} . (b) The same diagnostics for models trained on data from each scheme (median over three seeds). (c) Transfer error of the learned models against the rollout horizon TT . The model with the correct structure ( JAJ_{A} , black) has the third-lowest error at every horizon.
Figure 4: Semigroup error and fidelity diagnostics on the same learned time-steppers, coupled harmonic oscillators, median over three seeds. (a) Data from four solvers, one model class: the semigroup error is constant, while the energy diagnostic Da[H]D_{a}[H] and the structure diagnostic DjD_{j} separate explicit Euler by more than four orders of magnitude. (b) One data set, three model classes, of which only the first is a valid time-stepper: the semigroup error separates them.
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