Robust Ensemble Guidance for Scientific Inverse Problems

Zixiang Li; Wei Wang; Yunchao Wei; Yao Zhao; Yue Song

cs.LGOct 4, 2026

Robust Ensemble Guidance for Scientific Inverse Problems

Authors: Zixiang Li, Wei Wang, Yunchao Wei, Yao Zhao, Yue Song

Organizations: Beijing Jiaotong University · Tsinghua University

Abstract

Ensemble guidance combines pretrained diffusion priors with black-box forward models to solve inverse problems without differentiating through the physical simulator. However, observation coordinates with large predictive spread or extreme residuals can dominate the ensemble correction, degrading reconstruction accuracy. We show that two simple modifications, weighting and clipping, substantially improve this correction. Our method, Robust Ensemble Guidance (REG), uses ensemble predictive spread to balance observation scales and adaptively clips standardized residuals to limit the influence of extreme discrepancies. Both operations reuse existing particles and forward predictions, requiring no additional denoiser or forward-model evaluations. Under a local linear Gaussian model, we derive conditions for reduced one-step estimation risk, bound the influence of individual observation coordinates, and characterize when these benefits persist with finite ensembles. Experiments on Navier-Stokes inversion, black-hole imaging, and acoustic full-waveform inversion demonstrate improved reconstruction over the underlying ensemble solver. In particular, REG increases black-hole reconstruction PSNR by 6.2-8.2 dB across three observation regimes and reduces Navier-Stokes reconstruction error by 26.4% in a matched-budget comparison. These findings highlight the importance of observation heterogeneity and residual influence in designing reliable generative solvers for scientific inverse problems.

Figures & tables

Figure 1: Overview of Robust Ensemble Guidance. Forward predictions from a denoised particle ensemble determine coordinate weights and a shared clipping threshold. The resulting bounded residuals are transported back to the particle space to form the next correction.

Figure 2: Controlled tests of weighting, clipping, and the conditional one-step risk criterion.

Subsampling	$\times 2$			$\times 4$			$\times 8$
Method / $\sigma_{y}$	$0$	$1$	$2$	$0$	$1$	$2$	$0$	$1$	$2$
EKI	0.577 -0.6pt	0.609 -0.6pt	0.673 -0.6pt	0.579 -0.6pt	0.669 -0.6pt	0.805 -0.6pt	0.852 -0.6pt	0.940 -0.6pt	1.116 -0.6pt
DPS-fGSG	1.687 -0.6pt	1.612 -0.6pt	1.454 -0.6pt	1.203 -0.6pt	1.209 -0.6pt	1.200 -0.6pt	1.246 -0.6pt	1.221 -0.6pt	1.260 -0.6pt
DPS-cGSG	2.203 -0.6pt	2.117 -0.6pt	1.746 -0.6pt	1.175 -0.6pt	1.133 -0.6pt	1.114 -0.6pt	1.186 -0.6pt	1.204 -0.6pt	1.218 -0.6pt
DPG	0.325 -0.6pt	0.408 -0.6pt	0.466 -0.6pt	0.322 -0.6pt	0.361 -0.6pt	0.454 -0.6pt	0.596 -0.6pt	0.591 -0.6pt	0.846 -0.6pt
SCG	0.908 -0.6pt	0.928 -0.6pt	0.966 -0.6pt	0.869 -0.6pt	0.926 -0.6pt	0.929 -0.6pt	1.260 -0.6pt	1.284 -0.6pt	1.347 -0.6pt

Table 1: Navier–Stokes relative

\ell_{2}

error (

\downarrow

). Cells show mean above (s.d.); bold indicates the lowest mean in each column.

$J$	EnKG	REG	Change
512	$0.2752\pm 0.1088$	$\mathbf{0.2690}\pm 0.0890$	$-2.3\%$
1024	$0.2029\pm 0.1250$	$\mathbf{0.1493}\pm 0.0463$	$-26.4\%$
2048	$0.1156\pm 0.0385$	$\mathbf{0.1000}\pm 0.0314$	$-13.4\%$

Table 2: Matched ablation (

\times 2

\sigma_{y}=0

). Relative

\ell_{2}

error: mean

\pm

s.d.; bold: lowest mean per comparison. Changes are relative to EnKG at the same

J

Figure 3: Navier–Stokes reconstructions (top) and absolute errors (bottom) for the same test case, with shared color scales across methods.

Obs. time	$\mathbf{3\%}$				$\mathbf{10\%}$				$\mathbf{100\%}$
Method	PSNR $\uparrow$	Blur $\uparrow$	$\widetilde{\chi}^{2}_{cp}$	$\widetilde{\chi}^{2}_{\log ca}$	PSNR $\uparrow$	Blur $\uparrow$	$\widetilde{\chi}^{2}_{cp}$	$\widetilde{\chi}^{2}_{\log ca}$	PSNR $\uparrow$	Blur $\uparrow$	$\widetilde{\chi}^{2}_{cp}$	$\widetilde{\chi}^{2}_{\log ca}$
SMILI	18.51 -0.6pt	23.08 -0.6pt	1.478 -0.6pt	4.348 -0.6pt	20.85 -0.6pt	25.24 -0.6pt	1.209 -0.6pt	21.788 -0.6pt	22.67 -0.6pt	27.79 -0.6pt	1.878 -0.6pt	17.612 -0.6pt
EHT-Imaging	21.72 -0.6pt	25.66 -0.6pt	1.507 -0.6pt	1.695 -0.6pt	22.67 -0.6pt	26.66 -0.6pt	1.166 -0.6pt	1.240 -0.6pt	24.28 -0.6pt	28.57 -0.6pt	1.251 -0.6pt	1.259 -0.6pt
DPS	24.20 -0.6pt	30.83 -0.6pt	8.024 -0.6pt	5.007 -0.6pt	24.36 -0.6pt	30.79 -0.6pt	13.052 -0.6pt	6.614 -0.6pt	25.86 -0.6pt	32.94 -0.6pt	8.759 -0.6pt	5.456 -0.6pt
LGD	22.51 -0.6pt	28.50 -0.6pt	15.825 -0.6pt	12.862 -0.6pt	22.08 -0.6pt	27.48 -0.6pt	10.775 -0.6pt	13.375 -0.6pt	21.22 -0.6pt	26.06 -0.6pt	13.239 -0.6pt	13.233 -0.6pt
RED-diff	20.74 -0.6pt	26.10 -0.6pt	6.713 -0.6pt	9.128 -0.6pt	22.53 -0.6pt	27.67 -0.6pt	2.488 -0.6pt	4.916 -0.6pt	23.77 -0.6pt	29.13 -0.6pt	1.853 -0.6pt	2.050 -0.6pt

Table 3: Black-hole imaging over 100 images. Cells show mean above (s.d.); bold and underline mark the best and second-best PSNR and blurred PSNR (dB).

Figure 4: Selected black-hole reconstructions of one case at $3\%$ , $10\%$ , and $100\%$ observation time, with PSNR (dB). EnKG and REG share observations; other baselines retain native realizations.

Method	Relative $\ell_{2}\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$	Data misfit $\downarrow$
Adam	0.333 (0.086)	9.968 (2.083)	0.305 (0.120)	115.14 (52.10)
DPS	0.250 (0.154)	14.111 (6.820)	0.491 (0.161)	155.08 (92.17)
LGD	0.244 (0.024)	12.288 (0.889)	0.341 (0.047)	258.47 (26.40)
DiffPIR	0.204 (0.129)	16.113 (6.962)	0.554 (0.191)	88.53 (56.91)
DAPS	0.201 (0.103)	14.914 (4.184)	0.321 (0.067)	111.13 (71.33)
PnP-DM	0.259 (0.075)	11.983 (2.269)	0.431 (0.073)	308.84 (26.34)

Table 4: Acoustic FWI results. Mean (s.d.). Bold marks the best mean in each column.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: NS reconstructions and absolute errors at $\times 2$ , $\sigma_{y}=0$ . EnKG and REG use $J=2048$ ; numbers are relative $\ell_{2}$ errors of the displayed reconstructions.

Figure 6: REG reconstructions of NS id0 across downsampling factors $\times 2/4/8$ and noise levels $\sigma_{y}=0/1/2$ , with $J=2048$ . All panels share a vorticity scale.

Figure 7: Selected black-hole reconstructions: id0 at $100\%$ , id3 at $3\%$ , and id7 at $3\%$ observation time. Numbers are aligned PSNR (dB); evaluation follows Appendix B.3 .

Figure 8: Additional black-hole targets and REG reconstructions at $3\%$ and $100\%$ observation time, with $J=256$ . Numbers are aligned PSNR (dB).

Figure 9: FWI reconstructions and absolute errors for id2, id5, and id7, with $J=64$ . Velocity and error panels use shared color scales; numbers are relative $\ell_{2}$ errors.

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