Steering Diffusion Models to Rare Events
with Sequential Monte Carlo

Aavash Subedi¹ Tim Reichelt¹ Christopher Williams¹ Philip Stier¹ Yee Whye Teh¹ Saifuddin Syed²

¹University of Oxford ²University of British Columbia

Paper arXiv Code BibTeX

Particle trajectories run from Gaussian noise on the right to data on the left. A red band at the top marks the rare event. Guidance pushes particles towards it, larger dots carry more weight, and red crosses mark particles removed by resampling. — **DireSMC in one picture.** Particles start as noise (right) and are denoised towards data (left). A guidance term nudges them towards the rare event E (red band). Dot size shows each particle's importance weight; low-weight particles are dropped and high-weight ones copied (red crosses). The weights that end up inside E give the probability estimate p₀[E].

9× – 1413×

net speed-up over Monte Carlo on a climate emulator, growing with rarity

10⁻³ → 10⁻⁵

event probabilities recovered to within a few percent of the reference

No retraining

works at inference time with any pre-trained score model

The idea

In short

Rare events are where scientists most want answers: record heatwaves, failure modes, crop failures. Plain Monte Carlo needs roughly 1/p samples just to see one.
A diffusion model already walks from noise to data through a sequence of distributions. We use that walk as the bridge that classic rare-event methods have to build by hand.
DireSMC steers a population of particles towards the event and tracks importance weights as it goes. You get samples of the event and an unbiased estimate of its probability.

Why rare events are hard

Diffusion models now stand in for expensive simulators in weather and climate, molecular dynamics and materials design. They are cheap to sample, so the obvious way to estimate the probability of an event is to draw many samples and count.

That breaks down in the tails. The relative error of the Monte Carlo estimate is 1/√(N p), so an event with probability 10⁻⁵ needs about ten million samples for a 10% error bar. Most runs of a smaller budget see the event zero times.

Guidance can push a sampler towards the event, but on its own it changes the distribution you sample from. You get dramatic samples with no way to say how likely they are.

Method

Steering by changing the score

A diffusion model generates samples by integrating its score ∇_x log p_t from noise back to data, and it lands on the data distribution p₀. Change the score and you land somewhere else. We add the gradient of a reward r_t, chosen so that the new endpoint is the data distribution restricted to the rare event, p₀(x | E).

∇_x log p_t + ∇_x r_t integrate x₀ ~ p₀(x | E)

Getting that reward right, and correcting for the fact that we can only approximate it, is the rest of the method. DireSMC is a sequential Monte Carlo sampler that runs inside the denoising process. It needs three ingredients, and each one fixes a specific problem.

Soften, steer, weigh

Soften

An event like φ(x) ≥ τ has an all-or-nothing indicator with zero gradient almost everywhere. We swap it for a smooth Gaussian-CDF reward that still points towards the event from outside it.

Steer

At each denoising step we look ahead to the model's guess of the clean sample (Tweedie's formula) and add the reward's gradient to the drift. For linear observables this look-ahead is closed-form.

Weigh

Every particle carries a weight that corrects for the steering. When the weights degenerate we resample and refresh particles with an MCMC move. The weights inside E estimate p₀[E].

Left: the soft reward rises smoothly from 0 to 1 around the threshold tau, sharper for smaller delta. Right: the guidance strength is large below the threshold and falls off inside the event region. — **The soft reward.** Smaller widths δ approach the hard indicator (left). Outside the event the gradient stays large, so particles far from E are pushed hardest (right).

Results

Experiments

We test DireSMC on a Gaussian mixture where the exact answer is known, and on a pre-trained climate emulator.

Checking against an exact answer

We first test on a seven-mode Gaussian mixture where the true probability is known in closed form. Events of the form x₁ ≥ τ range from about 7 × 10⁻³ down to 1.4 × 10⁻⁵.

DireSMC recovers the reference with both the analytical and a learned score. Its coefficient of variation stays flat as the event gets rarer, while Monte Carlo at the same budget gets steadily worse. We also compare with the importance-sampling estimator of Manshausen et al. (2026), which uses the probability-flow ODE: DireSMC has lower variance at equal compute and a much smaller discretisation bias.

Three panels. Left: coefficient of variation falls with the number of particles; DireSMC with MADM rejuvenation is lowest. Middle: at equal compute, DireSMC's error stays flat across rarities while Monte Carlo grows. Right: the ODE baseline overestimates the probability by up to 2.5 times at 50 steps; DireSMC stays near 1. — **Gaussian mixture.** Left: error falls as particles are added; resampling and MCMC rejuvenation (MADM) help most. Middle: at equal compute, error stays flat as the event gets rarer. Right: with exact guidance, the ratio of estimate to truth across step counts K; ours sits at 1.

Heatwaves in a climate emulator

We take a pre-trained score-based climate emulator (Bouabid et al.), trained on CMIP6 output from MPI-ESM1-2-LR. It generates global monthly fields of temperature, precipitation, humidity and wind, conditioned on the month and the level of global warming ΔT.

We target the region hit by the June 2021 Pacific Northwest heatwave and ask how likely an extremely hot July is at ΔT = 1.5 K. The reference comes from about ten million Monte Carlo samples.

Four maps of July surface-temperature anomaly over North America. The unguided mean is mild. Three guided samples at increasing thresholds show strong warming centred on the Pacific Northwest box, with estimates 9.83e-4, 1.01e-4 and 1.06e-5 against references 1e-3, 1e-4 and 1e-5. — **Climbing the threshold.** The average July (left) and the highest-weight guided sample at three increasingly extreme thresholds. Estimates p̂ match the Monte Carlo references p*, with a variance reduction of 34× to 2722× at matched compute. The samples look physically sensible: more warming over land than ocean, and coherent patterns that extend beyond the target box.

Multivariate extremes

Compound events matter because the variables move together. We ask for Julys that are both extremely hot and extremely dry over the region, using a single reward on both variables. DireSMC recovers the joint probability and the dependence ratio d, which compares the joint probability with the product of the marginals.

Reference p*	p̂ / p*	CV	Dependence d*	d̂ / d*	Net speed-up
3.14 × 10⁻³	0.97 ± 0.02	0.07	31	1.04 ± 0.02	9×
2.12 × 10⁻⁴	0.99 ± 0.03	0.11	212	1.02 ± 0.03	47×
1.39 × 10⁻⁵	1.04 ± 0.02	0.08	1384	1.06 ± 0.06	1413×

Mean ± standard error over 16 seeds. Net speed-up is the Monte Carlo compute needed to match our variance, divided by our compute with the cost of guidance included.

The dependence grows sharply with rarity. At the rarest level a hot and dry July is over a thousand times more likely than treating heat and dryness as independent would suggest.

Two rows of maps for temperature and relative humidity. Top row: samples conditioned on heat only. Bottom row: conditioned on heat and dryness, showing a strong drying pattern over the Pacific Northwest box. — **Heat alone versus heat and drought.** Temperature (left) and relative humidity (right) for samples conditioned on a hot July only (top) and on a hot and dry July (bottom).

Climate attribution

The emulator is conditioned on global warming, so we can ask how the odds of one fixed extreme change as the planet heats up. We hold the threshold at a level that is about a 1-in-10,000 event at 1.2 K of warming, roughly the 2021 climate.

Its probability goes from about 10⁻⁶ in a pre-industrial climate to near certainty at 4 K. DireSMC tracks all six orders of magnitude with the same settings. At equal compute, Monte Carlo records no events at all in the coolest climates.

Estimated probability against warming from 0 to 4 kelvin on a log scale. DireSMC estimates rise smoothly from about 1e-6 to near 1. Monte Carlo records zero events at 0 and 0.5 kelvin and has huge error bars up to 1.2 kelvin. — Probability of the same heat extreme against global warming ΔT. The dotted line marks 2021.

This points towards using generative emulators for event attribution: asking how much more likely a given extreme has become because of warming.

Cite

BibTeX

@article{subedi2026diresmc,
  title   = {Steering Diffusion Models to Rare Events
             with Sequential Monte Carlo},
  author  = {Subedi, Aavash and Reichelt, Tim and Williams, Christopher
             and Stier, Philip and Teh, Yee Whye and Syed, Saifuddin},
  journal = {arXiv preprint},
  year    = {2026}
}