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Gappy probabilistic manifold decomposition for nonlinear field reconstruction
Qihan Feng, Jiaming Guo, Jiarun Meng, Dunhui Xiao
TL;DR
Extremely sparse and noisy measurements make high-dimensional field reconstruction difficult, especially when solution structures are strongly nonlinear. The paper proposes Gappy PMD to reconstruct fields on a learned nonlinear manifold, adds DPS to optimize sampling locations through differentiable interpolation, and derives an error analysis. Across three incompressible benchmarks, Gappy PMD substantially outperforms Gappy POD, while DPS further reduces error; the method has higher but millisecond-scale online cost.
Problem
Sparse measurements make direct field reconstruction underdetermined and sensitive to noise, while linear subspaces can require many modes for strongly nonlinear or convection-dominated fields.
Method
Gappy PMD combines SVD reduction with a probabilistic nonlinear residual manifold, and DPS directly optimizes sampling locations using smooth interpolation and reconstruction-error gradients.
Results
At equal reduced dimension and sampling budget, Gappy PMD achieves mean relative L2 errors one to two orders of magnitude smaller than Gappy POD across three benchmarks; DPS reduces error by a further 25% to 90%.
Takeaways & Limitations
Nonlinear manifold reconstruction and direct sampling optimization improve sparse field reconstruction accuracy across the evaluated flow problems, including under noisy measurements.
Takeaways & Limitations
Gappy PMD has higher online computational cost than Gappy POD because it requires nonlinear optimization and reconstruction from nonlinear coordinates, although the time per snapshot remains on the order of milliseconds.
Abstract
from arXiv · showhide
This paper proposes gappy probabilistic manifold decomposition (Gappy PMD), a nonlinear method for reconstructing high-dimensional fields from extremely sparse measurements. Gappy PMD reconstructs the field on the nonlinear manifold learned by probabilistic manifold decomposition (PMD). We further propose a differentiable point selection method for reduced-order model (ROM)-based field reconstruction (DPS). Using differentiable meshless interpolation within the ROM-based reconstruction framework, DPS makes the full-field reconstruction error differentiable with respect to the sampling locations and directly optimizes these locations. In addition, a theoretical error analysis for Gappy PMD is also given. It splits the squared reconstruction error into two orthogonal parts: one normal to the reconstruction manifold and the other induced by sparse sampling and observation noise. Under a stability condition on the sampling operator, this error vanishes with the PMD approximation error and the noise. The Gappy PMD is evaluated on three numerical test cases: flow past a cylinder, lid-driven cavity flow, and backward-facing step flow. For the same reduced dimension and sampling points, Gappy PMD attains mean relative $L^2$ errors one to two orders of magnitude below Gappy POD. Optimizing the sampling points with DPS further improves reconstruction accuracy and robustness.
1. Introduction
Sparse field reconstruction is underdetermined and becomes unreliable with extremely few noisy measurements, while linear methods struggle with strongly nonlinear solution structures. This paper addresses these limits with Gappy PMD, differentiable point selection, and a reconstruction-error analysis.
- Motivation: Sparse measurements make direct full-field reconstruction underdetermined and sensitive to measurement errors, while interpolation accuracy deteriorates sharply as measurements become extremely sparse.These limitations motivate methods that incorporate prior information about field structure.
- Motivation: Deep learning methods can reconstruct fields effectively but typically require extensive training datasets and encode the learned mapping implicitly in network parameters.The paper contrasts these requirements with snapshot-based reduced-order representations.
- Limits of existing methods: Gappy POD estimates modal coefficients from sparse measurements, but linear subspace approximation is inefficient for strongly nonlinear or convection-dominated phenomena.Slow Kolmogorov n-width decay forces linear methods to retain many modes for acceptable accuracy.
- Nonlinear reconstruction: Nonlinear methods such as Gappy DMAPs and Gappy AE address the geometric limitations of linear reconstruction by representing field data on low-dimensional nonlinear structures.Gappy DMAPs can maintain reconstruction accuracy with few or poorly located sampling points.
- Sampling design: Sampling-point selection critically affects reconstruction performance, motivating methods that optimize locations rather than relying only on proxy criteria.DEIM and QDEIM support Gappy POD, while differentiable strategies have also been developed for deep-learning reconstruction.
- Proposed approach: Gappy PMD combines linear SVD reduction with a probabilistic embedding of the nonlinear residual manifold, then estimates linear coefficients and manifold coordinates from sparse measurements.DPS makes sampling locations continuous through smooth interpolation so reconstruction error can be minimized directly; the paper also derives an error identity governed by manifold-normal error, noise, and sampling stability.
2. Gappy PMD and differentiable point selection
Gappy PMD combines a linear SVD representation with a nonlinear manifold model for reconstructing fields from sparse measurements, while DPS differentiably optimizes sampling locations.
- PMD representation: PMD represents fields with truncated SVD coefficients and models projection residuals on a nonlinear manifold.This addresses the slow Kolmogorov n-width decay that limits linear approximation for strongly nonlinear problems.
- Residual manifold coordinates: The residual manifold is learned from pairwise geodesic similarities, density-normalized transitions, and spectral coordinates that retain dominant geometric degrees of freedom.Geodesic distances are approximated on a nearest-neighbor graph, while eigenvalue weighting suppresses rapidly mixing modes.
- Lift mapping: The PMD lift reconstructs the state as the mean plus the linear component and a kernel-regressed nonlinear residual driven by joint manifold and linear coordinates.The joint input reduces ambiguity because the nonlinear embedding is not guaranteed to be injective.
- Gappy PMD reconstruction: Gappy PMD estimates linear and manifold coordinates from sparse measurements, using sampled residuals to extend the lift mapping from measured locations to unsampled nodes.The augmented lift mapping must be retrained when the sampling configuration changes.
- Gappy PMD reconstruction: The coordinate estimation is constrained to an expanded training-coordinate box, and experiments use q = d + 2 measurements so the residual system is moderately overdetermined.The box constraint keeps optimization within the region where the lift mapping was trained.
3. Theoretical analysis
The analysis establishes a local error identity and bound for Gappy PMD, decomposing reconstruction error into tangent-space and orthogonal components. Under stable sampling, the bound supports convergence as normal error, measurement noise, and stationary residual vanish, while recovering Gappy POD as a linear special case.
- Local error identity: The local error identity decomposes Gappy PMD reconstruction error into an orthogonal normal component and terms governed by tangent-space sampling and measurement noise.The decomposition is formulated locally around the reconstruction manifold and uses the tangent space at the recovered point.
- Stationary-point bound: At an interior stationary point, the stationary residual vanishes, yielding the bound ∥e∥2 ≤ eΛ(∥e⊥∥2 + ∥η∥2).The factor eΛ captures the effect of sampling stability on the local tangent space.
- QDEIM stability: QDEIM sampling bounds the tangent-space factor using its stability constant and local tangent-space misalignment, under the condition κqτq < 1.When τq = 0, the bound reduces to eΛ ≤ κq.
- Convergence: Under compactness, constant rank, uniform stability, and vanishing normal error, noise, and stationary residual, the reconstruction error converges to zero.The stated normal error is local at the recovered point and is not itself the global PMD approximation error.
- Gappy POD special case: Removing the nonlinear lift recovers the classical Gappy POD estimate, with orthogonal normal and tangent-space error terms and stationary factor eΛ = σmin(SΦ)−1.In this linear case, the tangent space is range(Φ), and the normal equations imply zero stationary residual.
4. Numerical results
Across three incompressible-flow benchmarks, Gappy PMD consistently outperforms Gappy POD at comparable sampling, while DPS further lowers errors and reduces error excursions. Noise increases reconstruction error progressively, but the method remains bounded under strong corruption.
- Computational cost: Gappy PMD requires higher online computational cost than Gappy POD, but reconstruction remains on the order of milliseconds per snapshot.The added cost arises from nonlinear optimization and reconstruction from nonlinear coordinates.
- Flow past a cylinder: Gappy PMD reproduces cylinder-flow vortex shedding with errors lower by orders of magnitude than Gappy POD using the same six sampling points.Gappy POD errors follow the wake, whereas Gappy PMD errors remain low except near passing vortex cores.
- Flow past a cylinder: DPS sampling lowers cylinder-flow error and removes the pronounced periodic spikes observed with QDEIM during vortex shedding.The resulting reconstruction is robust across the shedding cycle rather than accurate only at favorable phases.
- Noise robustness: At η% = 50, Gappy PMD with DPS maintains mean relative L2 error below 10%, with error growing approximately linearly beyond η% = 1.The noise contribution governs accuracy beyond η% = 1, but reconstruction does not break down under severe measurement corruption.
- Lid-driven cavity flow: In lid-driven cavity flow, Gappy PMD errors remain about two orders of magnitude below Gappy POD, while DPS reduces Gappy PMD mean error by more than half.DPS also reduces transient error peaks, indicating an additional benefit beyond the main gain from the PMD representation.
- Backward-facing step flow: For backward-facing step flow, DPS produces fewer high-error regions than QDEIM without increasing the number of sampling points.The two Gappy PMD reconstructions are visually indistinguishable, but their error fields differ in amplitude.
- Backward-facing step flow: In backward-facing step flow, Gappy PMD has about one order of magnitude lower mean relative L2 error than Gappy POD, while DPS reduces its mean error by about one quarter.These differences persist over the test interval rather than appearing only in instantaneous comparisons.
- Noise robustness: Measurement noise dominates the Gappy PMD error as soon as it is introduced, with approximately linear growth up to the largest tested noise level.The results show no accelerated degradation under strong corruption.
5. Conclusions
Gappy PMD reconstructs sparse-measured fields on a nonlinear PMD manifold, while DPS directly optimizes sampling locations against full-field reconstruction error. Theory characterizes approximation, sampling, and noise effects, and experiments show substantially lower errors than Gappy POD, with a fixed-configuration scope.
- Gappy PMD recovers full fields from extremely sparse measurements by solving for PMD coordinates and applying a nonlinear lift mapping.This nonlinear-manifold reconstruction captures solution structures that low-dimensional linear spaces may miss.
- DPS selects sampling points by directly minimizing full-field reconstruction error through differentiable interpolation and continuous-coordinate optimization.Optimized coordinates are ultimately snapped to distinct mesh nodes.
- The squared reconstruction error separates orthogonally into manifold approximation error and a term induced by sparse sampling and measurement noise.Under a sampling-operator stability condition, the error vanishes as PMD approximation error diminishes; the theory also recovers the classical Gappy POD estimate in the linear case.
- At equal reduced dimension and sampling budget, Gappy PMD achieves mean relative L2 errors one to two orders of magnitude smaller than Gappy POD across three incompressible benchmarks.Replacing QDEIM with DPS further reduces Gappy PMD error by 25% to 90%, while noisy-measurement error grows approximately linearly without breakdown under severe corruption.
- Gappy PMD is developed for a fixed configuration whose representation and sampling points are built offline from data at a single setting.Extending the method to varying Reynolds numbers or boundary conditions is identified as a natural direction.