Research

Layer 01 · 2004 – 2013

The hidden state

A system is being driven by dynamics too fast to resolve. You observe it noisily and only in part. What is it doing right now?

This is nonlinear filtering, and in multiscale systems the textbook answer is the wrong one. Filtering the full system is intractable; filtering a naively reduced system throws away the coupling that mattered. The work here established how to average and homogenize first, then filter — with the reduction carrying the fast scales' statistical effect rather than discarding it, and with convergence results that say when the reduced filter tracks the true one.

The systems ranged from impact oscillators with noise, through singularly perturbed stochastic hybrid systems, to power-system machine trajectories — and to the Lorenz '96 atmospheric model, which the lab has now been returning to for fifteen years.

Layer 02 · 2021 – 2024

The hidden function

Move the unknown up a level. It is no longer a trajectory but a function — a spectral density, an operator standing behind the measurement.

Recovering it means solving an ill-posed inverse problem, and ill-posed problems are decided by the prior. The work here replaced hand-designed regularizers with a learned prior: a normalizing flow, trained on physically admissible functions, which supplies an exact likelihood and so makes the posterior something you can actually sample rather than merely optimize against. Applied to optical spectra of correlated-electron materials, it recovers electron–boson spectral density functions that classical inversion cannot resolve.

Layer 03 · 2024 – present

The hidden model

Move it up once more. Now the unknown is the equation itself — the closure that stands in for physics the model cannot resolve, learned from data because no one can write it down.

And a second question arrives with it, which the first two layers never had to face. A learned closure can be accurate on every metric you checked and still be wrong about the thing you did not check. Weather forecasts can look healthy while the simulated climate quietly contracts. So the question is not only what is the closure, but which statistics does it preserve, and how would you know.

Why this matters now

Climate models cannot resolve clouds, and the unresolved processes are where the largest uncertainties live. The field's leading strategy is to replace expensive components with machine-learned surrogates — emulators that make thousand-member ensembles and kilometre-scale simulation affordable for the first time.

The bet is unproven exactly where it matters most. Emulators reproduce precipitation well and yet depart from robust physical behaviour in their radiative response; extremes and tails remain a conspicuous weak point; hybrid closures that behave offline can destabilize once coupled. Every one of these failures is a well-posed mathematical problem.

The quantities climate policy depends on — sensitivity, feedback, the frequency of extremes — are properties of a model's long-run invariant measure, not of its forecast skill. Certifying that learned components preserve those properties is the choke point, and it is where a small group doing careful mathematics can matter more than a large one doing engineering.

Methods we use

Stochastic dynamics and multiscale analysis

Averaging, homogenization, spectral budgets, invariant measures, Mori–Zwanzig memory.

Bayesian inference and inverse problems

Nonlinear filtering, data assimilation, learned priors, posterior sampling and coverage.

Generative and probabilistic machine learning

Normalizing flows, mixture density networks, and their use as measurement instruments rather than black boxes — an exact likelihood is a diagnostic, not just a loss.

Estimator discipline

Truth-versus-truth floors, perfect-model benchmarks, seed provenance. A surprising amount of published comparison is measuring its own sampling noise.