SURGE: Approximation-free Training Free Particle Filter for Diffusion Surrogate

Abstract: Diffusion-based generative models increasingly rely on inference-time guidance, adding a drift term or reweighting mixture of experts, to improve sample quality on task-specific objectives. However, most existing techniques require repeated score or gradient evaluations, introducing bias, high computational overhead, or both. We introduce \texttt{URGE}, Unbiased Resampling via Girsanov Estimation, a derivative-free inference-time scaling algorithm that performs path-wise importance reweighting via a Girsanov change of measure. Instead of computing gradient-based particle weights in previous work, \texttt{URGE} attaches a simple multiplicative weight to each simulated trajectory and periodically resamples. No score, no Hessian, and no PDE evaluation is required. We establish an equivalence between path-wise and particle-wise SMC: the Girsanov path weight admits a backward conditional expectation that recovers the previous particle-level weights, guaranteeing that both schemes produce the same unbiased terminal law. Empirically, \texttt{URGE} outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks, achieving better generation quality, while being significantly simpler to implement and fully gradient-free.

• The paper presents SURGE, a novel particle filtering method that achieves unbiased posterior inference in diffusion-based data assimilation.
• It combines guided sampling with Girsanov correction to compute exact path-wise importance weights, ensuring unbiased results despite heuristic drift adjustments.
• Experiments on Lorenz-63, Navier-Stokes, and real-world weather forecasting validate SURGE's accuracy and efficiency in high-dimensional, complex systems.

SURGE: Approximation-Free Inference-Time Particle Filtering for Diffusion Surrogates

Problem Setting and Motivation

Data assimilation (DA) addresses sequential inference of a dynamical system’s latent state from noisy and partial observations, a scenario intrinsic to fields such as meteorology, oceanography, and fluid dynamics, where the governing dynamics are often nonlinear, high-dimensional, and stochastic. The advent of data-driven "digital twins" built via powerful generative models—specifically conditional diffusion models—has led to new DA challenges. When the transition operator of the latent process is simulated by a diffusion surrogate rather than explicit time-stepping solvers, fusing model forecasts with physical observations to yield a physically consistent, computationally tractable posterior remains non-trivial.

Traditional methods for assimilating observations into diffusion models often use guidance techniques based on likelihood terms, but these are fundamentally biased unless the exact Doob’s $h$-transform is available, which is typically intractable for complex, high-dimensional systems. This introduces systematic bias and can impair posterior consistency, especially in settings where the physical system is chaotic, high-dimensional, or where observability is sparse and noisy. The central technical question is how to design a DA framework that fuses diffusion surrogate predictions and observations in an unbiased manner, without retraining or heuristic adjustments to the generative model.

Methodology: SURGE Particle Filtering Framework

The core contribution of the paper is SURGE (Sequential Unbiased Resampling via Girsanov Estimation), an inference-time, training-free, approximation-free data assimilation method combining a pretrained diffusion surrogate with sequential observational data using a particle filtering approach operating on the path measure of the internal diffusion SDEs.

Key elements of the methodology:

• Conditional Diffusion Surrogacy: The transition density $p(x_{t+1}|x_t)$ is approximated by a conditional diffusion model, parameterized as an SDE over an internal time variable, providing stochastic, high-dimensional prior dynamics.
• Guided Proposals and the Girsanov Correction: When new observations are available, guided sampling in the diffusion process injects observation-consistency through ad hoc or heuristic drift corrections. However, unless the guidance equals the true Doob $h$-transform, resulting proposals are biased. SURGE corrects for this by computing path-wise importance weights via Girsanov’s theorem—quantifying the exact Radon-Nikodym derivative between the reference (unguided) diffusion and the guided proposal measure—ensuring unbiased targeting of the true filtering posterior.
• Sequential Monte Carlo (Particle Filtering) on Trajectories: An ensemble of trajectories (particles) is evolved according to the guided diffusion SDE. At each time step, path-wise importance weights are updated, and systematic resampling is performed based on the effective sample size, focusing computational effort on high-likelihood trajectories.
• Progressive Likelihood Incorporation: To counteract weight degeneracy, which is exacerbated in high-dimensional and long-horizon contexts, SURGE incrementally applies the observational likelihood along trajectory intervals, rather than as a single, terminal update. This stabilizes the importance weights by distributing the likelihood across diffusion steps, reducing variance and enhancing resampling efficacy.
• Training-Free, Plug-and-Play Application: SURGE operates exclusively at inference, making no modifications to the pretrained diffusion surrogate itself. It can be applied post hoc to any diffusion-based DA method without retraining, enhancing both classical and modern baselines.

Theoretical Analysis

Theoretical guarantees are provided for the correctness and approximation-freeness of the SURGE filter:

• Exactness via Importance Sampling: In the large particle limit, SURGE’s self-normalized estimator converges to the true filtering posterior, as established through the Radon-Nikodym derivative (given by the Girsanov change of measure) applied to the path measure.
• No Systematic Bias from Guidance: The approximation-free property is rigorously formalized: inadequacies in the guided proposal affect only the variance and efficiency of the estimator but do not introduce bias into the posterior, provided sufficient particles and correct weight computation.
• Progressive Likelihood and Stability: The telescoping product form for gradual likelihood incorporation retains exactness of the overall posterior and is equivalent to terminal likelihood weighting in the zero-step-size limit.

Experimental Evaluation

Extensive experiments validate SURGE across a spectrum of DA benchmarks:

• Lorenz-63 System: In this chaotic, low-dimensional setting, SURGE significantly improves posterior state estimates under strong partial observability and high measurement noise, consistently lowering RMSE and Wasserstein-1 distance compared to bootstrap particle filter (BPF), EnKF, score-based DA (SDA), and FlowDAS baselines.
• High-dimensional Navier-Stokes Flow: For 2D, forced incompressible Navier-Stokes on a $128^2$ grid (with super-resolution and extreme sparsity in observation), SURGE produces more physically accurate posterior evolution, maintaining lower pixel-level RMSE and kinetic energy spectrum relative error over long autoregressive prediction horizons.
• Real-world Weather Forecasting (SEVIR): When applied to deep-learning surrogates for precipitation nowcasting under severe data sparsity, SURGE yields higher CSI and lower RMSE across both moderate and extreme event thresholds compared to all evaluated baselines, including FlowDAS with and without guidance.

In all scenarios, SURGE consistently demonstrates quantifiable performance improvements in both filter accuracy and physical fidelity, confirming its stability and scalability in high-dimensional, long-horizon DA regimes.

Critical Analysis and Practical Implications

Strong empirical results: Across all tasks, SURGE results in statistically significant improvements over classical and learned baselines. The largest performance gains are observed in settings with very sparse, partial observation and strong stochasticity in the true dynamics, where naive guidance or heuristic posterior corrections typically yield biased or unstable estimates.

Approximation-free and post hoc applicability: By correcting guided proposals at the path measure level via Girsanov’s theorem, SURGE eliminates the reliance on tractable Doob’s $h$-transforms or backward Kolmogorov solutions. Its design as a plug-and-play inference module enables it to enhance a wide array of diffusion-based DA systems without costly retraining.

Limitations and challenges:

• Dependency on surrogate quality: The method’s efficacy is sensitive to the quality and stability of the pretrained diffusion surrogate. If the surrogate is miscalibrated or degenerate—such as in regions poorly covered by training data—the particle filtering is susceptible to weight degeneracy, overshooting, or selection bias.
• Particle filter scaling: While progressive likelihood helps, particle filtering inherently suffers from exponential scaling in the state dimension. Empirically, SURGE maintains high effective sample size relative to vanilla filters, but its efficiency in even higher-dimensional or longer-horizon scenarios remains an active research challenge.

Contradictory to prior claims: The work strongly contests the sufficiency of simple likelihood guidance or flow-matching techniques for unbiased posterior inference in diffusion DA. The analysis and experiments show that these heuristics generally introduce bias, only correctable via exact path-level importance weighting, as deployed by SURGE.

Theoretical and Future Directions

SURGE advances theoretical understanding by providing a concrete, algorithmically tractable instantiation of path-space, approximation-free Bayesian data assimilation for diffusion surrogates. The separation between proposal efficiency (qualitative surrogate and guidance design) and posterior consistency (path-weight correction) establishes a robust paradigm for test-time DA in complex systems.

Potential directions for future work include:

• Adaptive guidance/reward design: Dynamically tuning guidance/control potentials and reward mechanisms to improve proposal efficiency, especially in regimes with model misspecification or sharp transitions.
• Variance reduction and scalable sampling: Leveraging novel SMC or resampling algorithms tailored to diffusion-generated proposals to mitigate degeneracy in even larger latent spaces.
• Multimodal and multi-physics assimilation: Extending the methodology to multi-scale or multi-physics settings, where observations arise from physically distinct modalities requiring hierarchical or coupled surrogates.
• Real-time systems and uncertainty quantification: Applying SURGE to real-time forecasting applications where both speed and uncertainty quantification are critical.

Conclusion

SURGE provides a theoretically rigorous, training-free, inference-time particle filtering framework for data assimilation with diffusion model surrogates. By correcting the bias of guided diffusion sampling at the path level via Girsanov importance weighting and progressive likelihood incorporation, SURGE enables robust, approximation-free posterior inference in high-dimensional, nonlinear systems encountered in scientific and engineering domains. Empirical results confirm substantial improvements over state-of-the-art baselines, especially in settings characterized by strong stochasticity, sparse and noisy observations, and demanding physical priors. The proposed framework represents a significant step toward principled, scalable assimilation of real-world data into generative digital twins across scientific disciplines.