Amortized Localized Posterior Inference for Model Calibration with Application to an Ocean Turbulence Closure

Introduction

Calibrating complex physical models often runs into two intertwined challenges: model discrepancy—where a single set of parameters cannot capture behavior across varying conditions—and uncertainty in both observations and the contexts that drive the model. Traditional global Bayesian calibration treats all data equally, which can mask local compensations and inflate predictive errors. A recent approach, amortized localized posterior inference, tackles these issues by constructing a family of posteriors that give higher weight to observations most relevant to a target context, then learning this family with a conditional normalizing flow.

Localized Posteriors via Context Weighting

The method begins by defining a context space (e.g., atmospheric forcing for an ocean model). For any target context \(c\), observation likelihoods are re‑weighted with a Gaussian kernel that measures proximity in context space. This kernel can be tuned by inflating its covariance to reflect uncertainty about the context itself, and it includes safeguards such as reflection at domain boundaries and optional corrections for uneven sampling of contexts. The result is a localized posterior that emphasizes data most informative for the chosen context while still respecting the overall probabilistic framework.

Amortized Variational Inference with Normalizing Flows

Rather than solving a separate inference problem for each context, the approach leverages amortized variational inference. A conditional normalizing flow—an expressive, invertible neural network—takes the target context as input and outputs an approximation of the corresponding posterior distribution over calibration parameters. Training proceeds by minimizing a KL divergence across many sampled contexts, effectively teaching the flow to map any context to its appropriate posterior in a single forward pass.

From Local to Global: Marginalized Parameter Distributions

Once the family of localized posteriors is learned, they can be integrated over a prescribed context measure (e.g., the empirical distribution of atmospheric forcings) to obtain a marginalized parameter distribution. This distribution captures the full range of plausible calibrations across the entire context domain, offering a richer uncertainty quantification than a single global posterior.

Empirical Evaluation

Two experiments illustrate the benefits:

1. Twin experiment with the Branin‑Hoo function – Synthetic data were generated under both uniform and skewed context distributions. Comparing global calibration to the localized approach showed a clear reduction in aggregate predictive error, especially when contexts were unevenly sampled.
2. Application to a k‑ε turbulence closure in a single‑column ocean model – Here, atmospheric surface forcing served as the context. Localized posterior calibration yielded lower temperature and zonal‑velocity errors than both deterministic calibration and the global Bayesian posterior, demonstrating practical gains for a realistic geophysical system.

Both setups were designed to admit local compensation, meaning that different parameter tweaks could correct errors in specific contexts. The localized method successfully exploited this property, whereas the global posterior averaged over compensations and performed worse.

Practical Takeaways

• Context‑aware weighting helps isolate the most relevant data for each calibration scenario, mitigating the impact of model discrepancy.
• Amortized inference with normalizing flows makes the approach computationally feasible, enabling rapid posterior queries for new contexts.
• Marginalization over the context distribution provides a comprehensive uncertainty picture, useful for downstream decision‑making.
• The method is particularly advantageous when context sampling is non‑uniform or when local model errors dominate.

Conclusion

Amortized localized posterior inference offers a principled, scalable way to calibrate complex models under heterogeneous conditions. By blending context‑dependent likelihood weighting with modern amortized inference techniques, it delivers more accurate and reliable parameter estimates, as evidenced by improvements in synthetic benchmarks and a real‑world ocean turbulence closure. Researchers and practitioners facing calibration challenges across variable environments should consider this localized, amortized framework as a robust alternative to traditional global Bayesian methods.