Weak-DMD: A Galerkin approach to the problem of noise in the Dynamic Mode Decomposition algorithm

Abstract: Dynamic Mode Decomposition (DMD) is a data-driven method for approximating the spatiotemporal modes of a system. The eigenvectors and eigenvalues of the system are approximated from a series of time-snapshots of the state variables. The standard formulation of DMD is subject to strict assumptions concerning the time-spacing of the snapshots and is biased by measurement noise. Variations on the method have been developed to address these shortcomings, but the problem is still open. Motivated by the effectiveness of Galerkin methods in the field of model discovery, a weak formulation of DMD is presented, weak-DMD. Weak-DMD precludes timestep considerations and also filters noise. Results for two nuclear engineering applications and the flow of fluid past a cylinder are given and compared with a state of the art DMD algorithm.

• The paper introduces weak-DMD, a Galerkin formulation that removes DMD’s equal-timestep requirement and reduces noise through projection onto a compactly supported trial basis.
• The method adds noise primarily as an additive residual, outperforming optimized-DMD in most synthetic-noise tests and correctly identifying supercritical reactor growth when optimized-DMD consistently predicted decay.
• The results show substantially lower cylinder-flow forecast errors and reliable reactor estimates, but manual basis selection, computational cost, and unresolved spectral convergence limit immediate adoption.

Overview

Dynamic Mode Decomposition (DMD) approximates the eigenvalues and spatial modes of a linear operator governing a system's evolution from time-snapshot data, and is widely used for turbulent flow analysis, neutron transport eigenvalue calculation, and dimensionality reduction in control. Standard DMD requires equally spaced snapshots and is biased by measurement noise; existing remedies each carry restrictions — total-DMD (tDMD) addresses bias but assumes deterministic dynamics, subspace-DMD (sDMD) makes structural assumptions on the noise, and optimized-DMD, while flexible and noise-resilient, solves a nonlinear least-squares problem without convergence guarantees (2604.14350).

The paper under review introduces weak-DMD, a Galerkin (weak) formulation of DMD inspired by weak-SINDY in model discovery. The method lifts the equal-timestep requirement entirely, imposes no assumptions on noise structure or on the underlying dynamics beyond linearity of the governing ODE system, and reduces noise through projection onto a trial basis. Results are presented for two neutronics criticality problems — one with synthetic Gaussian noise and one with intrinsic Monte Carlo noise — and for turbulent flow past a cylinder, benchmarked against optimized-DMD from PyDMD with bagging.

Method derivation

Weak-DMD begins from noisy measurements $$\boldsymbol{x}_n = \boldsymbol{y}(t_n) + \boldsymbol{\epsilon}_n$$ assumed to be governed by a linear ODE system $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$. The data are projected onto a non-orthogonal trial basis $\{\psi_j\}_{j=1}^J$, with expansion coefficients obtained by solving a Gram-matrix system via least squares (the Gram matrix becomes ill-conditioned as the basis grows). The central assumption is that this projection yields a substantially de-noised representation of the state, allowing substitution into the weak form of the ODE: multiplying by compactly supported test functions $\phi_i$, integrating by parts over a time window $[t_1,t_2]$, and collecting terms produces

$$\boldsymbol{Y}^+ = \boldsymbol{A}\,\boldsymbol{Y}^-,$$

where $\boldsymbol{Y}^-$ contains inner products of the trial expansion against the test functions and $\boldsymbol{Y}^+$ additionally carries boundary correction terms and inner products against test-function derivatives. Notably, the trial basis is not required to vanish at the window boundaries; supports may extend outside the window (even into negative time), which better represents dynamics at the extrema at the cost of an explicit boundary term.

A thin SVD of $\boldsymbol{Y}^-$ then yields the reduced operator $$\widetilde{\boldsymbol{A}} = \boldsymbol{L}^T\boldsymbol{Y}^+\boldsymbol{R}\boldsymbol{S}^{-1}$$, whose eigenpairs approximate those of $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$0 by the standard DMD argument. Future prediction uses an implicit step with $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$1 projected back to full space, and spatial modes follow from the eigenvectors of $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$2 combined with $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$3.

A worked example on a damped oscillator with known eigenvalues $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$4 demonstrates the noise structure analytically: because zero-mean noise integrates to zero against the basis up to sampling error $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$5, the reconstructed operator satisfies $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$6, i.e., residual noise enters additively rather than multiplicatively as in standard DMD. This is the formal basis for the method's noise filtering claim. The same example also shows that even with optimal bases, a sufficiently long integration window is required: with $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$7 the computed eigenvalues are badly wrong ($d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$8), while $d\boldsymbol{y}/dt = \boldsymbol{A}\boldsymbol{y}$9 recovers them to five digits.

Implementation choices

Both test and trial spaces use compactly supported polynomial bases of the form used in weak-SINDY, parameterized by support endpoints $\{\psi_j\}_{j=1}^J$0 and polynomial order $\{\psi_j\}_{j=1}^J$1. Trial-basis construction is currently hand-tuned: the user specifies the number of functions per level and overlap percentages on a uniform center grid. The authors state plainly that automating this selection is deferred to future work. Two empirical rules emerged: enlarging the trial space without bound interpolates the noise (so trial spaces must stay modest), while the test space can be grown freely since it does not interpolate data. The SVD rank is chosen by an energy criterion retaining the smallest $\{\psi_j\}_{j=1}^J$2 such that cumulative singular-value energy exceeds a threshold $\{\psi_j\}_{j=1}^J$3.

Results

Twelve-group criticality problem

Synthetic data from a 12-group prompt-neutron sphere model (subcritical $\{\psi_j\}_{j=1}^J$4 cm, supercritical $\{\psi_j\}_{j=1}^J$5 cm) with analytically known eigenvalues were analyzed over $\{\psi_j\}_{j=1}^J$6. Without added noise, weak-DMD eigenvalues converge monotonically toward the benchmark as the test space grows (43 trial bases, $\{\psi_j\}_{j=1}^J$7, $\{\psi_j\}_{j=1}^J$8), outperforming optimized-DMD with SVD rank 4 and 10 bagging trials. With Gaussian noise of magnitude 15% of the local population and variance $\{\psi_j\}_{j=1}^J$9, convergence becomes more erratic, and the trial space had to be reduced from 43 to 14 bases with lower retained energy to avoid overfitting. Weak-DMD outperformed optimized-DMD for all but the second subcritical eigenvalue under noise. The authors explicitly caution against interpreting this as superiority, noting an expert optimized-DMD user could tune parameters more favorably.

Monte Carlo transport benchmark

The Kornreich–Parsons multi-region slab reactor benchmark was solved with MCDC using 182 spatial points, 196 angular directions, and 420 unevenly spaced timesteps at $\phi_i$0, $\phi_i$1, and $\phi_i$2 particles — noise here is intrinsic statistical uncertainty, not synthetic. Benchmark eigenvalues are $\phi_i$3 and $\phi_i$4. Key findings:

Method	Particles	$\phi_i$5 estimate	Error 1
weak-DMD	$\phi_i$6	0.0251	0.0125
weak-DMD	$\phi_i$7	0.1483	0.1107
weak-DMD	$\phi_i$8	0.0154	0.0222
optimized-DMD	all counts	negative	0.053–0.575
VDMD	$\phi_i$9/$[t_1,t_2]$0	≈0.01/0.018	0.028/0.020

The most consequential result is qualitative but strong: optimized-DMD returned a negative dominant eigenvalue for every particle count, incorrectly characterizing a supercritical assembly, whereas weak-DMD correctly identified positive growth in all cases. Weak-DMD also achieved the lowest dominant-mode error at $[t_1,t_2]$1 particles (0.0125 versus 0.0946 for VDMD and 0.0529 for optimized-DMD). VDMD with aggressive singular-value truncation ($[t_1,t_2]$2) beat both methods on the dominant mode at higher particle counts, indicating weak-DMD is not uniformly best. Weak-DMD benefited here from sparse snapshots, peaked bases ($[t_1,t_2]$3), and low energy tolerance ($[t_1,t_2]$4).

Flow past a cylinder

For OpenFOAM velocity data at Reynolds numbers 300–1000, no analytic eigenvalues exist, so performance was measured by averaged relative forecast error over $[t_1,t_2]$5 s after training on $[t_1,t_2]$6 s. Weak-DMD forecast errors were orders of magnitude smaller than optimized-DMD across the Reynolds range, and its leading modes appeared more turbulent in character while optimized-DMD's dominant modes were smoother and structured. Optimized-DMD was faster overall, but subtracting the forecast step narrowed the gap considerably — the authors identify their implicit forecast propagation as the current computational bottleneck.

Limitations and open questions

The paper is candid about several constraints. The trial-basis construction is manual and hand-tuned per problem; automation informed by data characteristics (Fourier content, estimated noise properties) is identified as a significant open challenge. The de-noising claim rests on the assumption that trial projection recovers the true state, supported empirically by reconstruction plots rather than proven. Theoretically, the relationship between weak-DMD spectra and Koopman spectra is unclarified, and no analysis of nonlinear systems is provided — despite the cylinder results suggesting empirical robustness to state stochasticity. Whether the algorithm converges to a fixed spectrum as the test space grows infinitely remains open, as does whether alternative trial/test basis families would improve performance. The rules of thumb (larger test than trial space; fewer retained singular values when snapshots are few) are empirical observations consistent with weak-SINDY findings but not established theoretically.

Conclusion

Weak-DMD reformulates DMD as a Galerkin problem, simultaneously removing snapshot-spacing restrictions and filtering measurement noise through trial-space projection, with residual noise entering the reconstructed operator only additively. Against optimized-DMD it delivered comparable accuracy on a synthetic-noise problem with known eigenvalues, correctly classified a supercritical Monte Carlo slab reactor where optimized-DMD failed qualitatively, and produced far more stable forecasts on turbulent cylinder wake data — at higher cost, driven by an inefficient forecast step. The evidence supports weak-DMD as a credible alternative for noisy, unevenly spaced data, contingent on future work automating basis selection and establishing spectral convergence theory.