Foresight: Learning Twisted SMC for Rare Events from Reversed Relaxations

Ziyi Fan; Yuxuan Qiu; Haibo Wang; Wentao Cui; Chen Fang

cs.LGOct 31, 2026

Foresight: Learning Twisted SMC for Rare Events from Reversed Relaxations

Authors: Ziyi Fan, Yuxuan Qiu, Haibo Wang, Wentao Cui, Chen Fang

Organizations: Key Laboratory of Organ Regeneration and Reconstruction, Chinese Academy of Sciences, Beijing, China · Institute of Industrial Science, The University of Tokyo, Tokyo, Japan · Westlake University, Hangzhou, China · Beijing Life Science Academy, Beijing, China

Abstract

We consider the probability that a reversible Markov chain started in a set $A$ is in a rare set $B$ after a fixed horizon, a quantity often needed for many start laws, down to single states. Twisted sequential Monte Carlo (SMC) estimates such probabilities without bias, with small variance when its proposal and twist approximate the dynamics conditioned on reaching $B$ , but these components are usually learned from forward simulations, which contain almost no successful paths when the event is rare. We introduce Foresight-SMC, which learns both components from the target side: under reversibility, relaxations started in $B$ that return to $A$ are, read backwards, exact samples of the conditioned paths, and their number is governed by the return probability rather than by the rare probability. Extending this reversal to the recorded propose--accept decisions of a Metropolis-adjusted sampler, we fit a controlled Langevin proposal with exact likelihood ratios by maximum likelihood, together with a classifier twist; the resulting estimator is unbiased for any start law. A variance analysis shows that resampling reduces the variance when the importance weights degenerate, as they do over long horizons. On an Allen--Cahn field, the learned proposal alone suffices, and one trained model is more accurate than splitting at equal cost. On alanine tripeptide, resampling with the learned twist is necessary, and Foresight-SMC needs less computation per query than adaptive multilevel splitting on localized start sets.

Figures & tables

Figure 1: Overview of Foresight-SMC. (a) Relaxations start from $Y_{0}\sim\mu_{B}$ ; those that end in $A$ , read backwards, are samples of $P^{\star}_{\mu_{A}}$ (Proposition 3.1 ) and train the proposal and the twist. (b) For a query, particles move from $A$ (blue) toward $B$ (red) under the controlled kernel $K^{\theta}$ and are resampled at block ends. Inset: one step, in which the reference Gaussian proposal (dashed) is shifted by $2\tau\nabla_{x}g_{\theta}$ and the acceptance is tilted by $e^{g_{\theta}}$ ; crosses mark rejected proposals. (c) Twisted SMC with proposal $K^{\theta}$ and twist $\widehat{H}_{k}$ returns an unbiased estimate of $p(\nu,T)$ ; with $L=1$ it reduces to importance sampling.

Figure 2: Allen–Cahn field, one trained model for all queries. (a) Absolute log error of the

60

point queries at equal total cost. (b) Cumulative evaluations of

U

and

\nabla U

: Foresight-SMC pays a one-off cost, whereas AMS and FFS, run to

20\,\%

RSE, pay again for every query.

Method	Cost	$E_{\log}$ (nat) $\downarrow$	In 2 SE
Foresight-SMC, $L=1$	1.32	$0.070\pm 0.002$	56–58
Foresight-SMC, 50-step blocks	1.31	$0.119\pm 0.027$	52–60
Reference-proposal SMC	1.31	$\infty$	–
AMS, equal cost	1.29	$0.296$	48
AMS, to $20\,\%$ RSE	4.02	$0.175$	51
FFS, to $20\,\%$ RSE	127	$0.059$	57

Table 1: Allen–Cahn field. Cost:

10^{9}

evaluations for all

66

queries, including the one-off cost of Foresight-SMC.

E_{\log}

: over the

60

point queries; Foresight-SMC gives mean

\pm

SE over three training seeds. In 2 SE: point queries within two standard errors of the reference.

\infty

: zero estimates on some queries. Further baselines: Appendix E .

Figure 3: Alanine tripeptide, eight broad queries. (a) Effective sample size of the weights along the horizon (median and 10–90 % band over queries), over the whole path for

L=1

and per block otherwise. (b) Mean number of residues with

\phi_{i}>0

among the resampled particles.

E_{\log}

of each variant: Table 2 .

Figure 4: Alanine tripeptide, four narrow start sets, against the fraction

r_{S}

of relaxations that end in each set. (a) Work-normalized variance; the cost of reverse counting grows as

1/r_{S}

(dashed). (b) Absolute log error against the reverse-counting reference. RC: reaction coordinate.

	Broad (8 sets)		Narrow (4 sets)
Method	Cost	$E_{\log}\downarrow$	$E_{\log}\downarrow$	WNV ratio $\downarrow$
Foresight-SMC, twist:
tempered classifier	6.6	$0.13\pm 0.03$	$0.15$	$1$
classifier	6.6	$0.18\pm 0.01$	$0.16$	0.45–5.8
$e^{g_{\theta}}$	6.6	$0.47$	$0.34$	3.3–30
Foresight-SMC, $L=1$	6.6	$1.26$	$1.03$	3.3–17 †

Table 2: Alanine tripeptide. Cost:

10^{7}

evaluations per broad query, excluding one-off costs; on the narrow sets, costs are similar except for learned-RC AMS (Appendix F.2 ).

E_{\log}

: mean absolute log error (nat);

\pm

: SE over three training seeds. WNV ratio: work-normalized variance relative to the first row, range over the four narrow sets. The plug-in gives no error bar; reverse counting is the narrow-set reference. † : effective sample size below

12

, so the WNV is underestimated. Per-set values: Tables F.2 and F.1 .

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure D.1 : Number of blocks

L

at a fixed number of evaluations, mean over four repetitions (bars: SE). (a) Mean absolute log error: alanine tripeptide, two broad queries, tempered classifier twist; Allen–Cahn field, six point queries at

T=500

, twist

e^{g_{\theta}}

. (b) Fraction of particles in

B

T

. Values differ from Tables 1 and F.1 , which average over all queries.

	Allen–Cahn field	Alanine tripeptide
MALA step	$\tau=0.006$ , $T\in\{250,500\}$	$\tau=10^{-3}$ on unwrapped increments, $T=16000$
Sampling $\mu_{B}$	MALA restricted to $B$ : 1024 chains, burn-in $6\times 10^{5}$ steps, 128 draws per chain	exact: rejection sampling of $\phi$ , then $\psi\sim\pi(\psi\mid\phi)$
Relaxation pool	131072 paths of 500 steps ( $74\,\%$ in $A$ at $t=500$ ); train/validation/test $90/5/5$ , split by chain	32768 paths ( $r_{B}=0.33$ ); held-out pool of 4096 for checkpoint selection
Records used	all ages, both horizons	128 per returning path, ages uniform
$g_{\theta}$	1-D CNN, 4 layers $\times$ 64 channels, kernel 7; remaining steps enter through Fourier features	MLP, 4 layers $\times$ 256, SiLU; inputs $\cos$ / $\sin$ of the angles, $U$ , Fourier features of the remaining steps, $\mathbf{1}_{B}$
Proposal	two-component mixture, Appendix B.1	same

Appendix

Table E.1 : Settings for the Allen–Cahn field and alanine tripeptide.

Figure E.1 : Allen–Cahn field: reference dynamics as space–time plots of the field $u_{i}$ (color; red: $u_{i}<0$ , the unfavored phase) over the sites $i$ and the steps $t$ . (a) From a point start in $A$ without a negative site, no droplet forms within $T=500$ . (b) A relaxation from $\mu_{B}$ : the droplet shrinks and vanishes; read backwards, the relaxation is a training path.

Seed	KL, $T=500$	KL, $T=250$	$E_{\log}$ , $L=1$	$E_{\log}$ , 50-step blocks
1	$3.24\pm 0.07$	$5.67\pm 0.17$	$0.073$ (58)	$0.065$ (60)
2	$3.27\pm 0.07$	$5.68\pm 0.18$	$0.067$ (56)	$0.146$ (55)
3	$3.26\pm 0.08$	$5.67\pm 0.17$	$0.071$ (56)	$0.146$ (52)

Appendix

Table E.2 : Allen–Cahn field: training seeds. KL is the held-out path KL for the start law

\mu_{A}

; for comparison,

-\log p

22.3

nat at

T=500

and

23.3

nat at

T=250

E_{\log}

over the 60 point queries, with the number of queries within two standard errors in parentheses. Seed 1 has the best validation objective.

$T$	Start	Foresight-SMC	Reference	TV
500	$\mu_{A}$	.397 / .436 / .167	.394 / .381 / .225	0.058
	0	.423 / .285 / .292	.396 / .316 / .288	0.031
	1	.347 / .259 / .394	.408 / .228 / .364	0.060
	2	.009 / .005 / .986	.010 / .010 / .981	0.005
	3, interior	.001 / .002 / .998	.000 / .003 / .997	0.001
	3, left end	.990 / .002 / .008	.991 / .000 / .009	0.001

Appendix

Table E.3 : Allen–Cahn field: channel shares (left end / right end / interior) against the reference, and their total variation (TV). Starts:

\mu_{A}

(reference: reverse counting) and one point start of each stratum, labeled by its number of negative sites (reference: AMS).

Figure E.2 : Allen–Cahn field: paths generated by Foresight-SMC with

L=1

from the droplet-free point start, one per channel (rows: interior, left end, right end), each the path of median importance weight in its channel. Columns show the field

u_{i}

t=0,100,\dots,500

; the dashed line is

u=0

and the shaded area marks negative sites.

Figure F.1 : Alanine tripeptide: start sets on the

(\phi_{i},\psi_{i})

plane of each residue, over the marginal free energy (saturated above

40

kJ mol -1 ). The box is the range of

\phi_{i}

A

; dashed lines mark the band

\psi_{i}\in[-120^{\circ},50^{\circ})

, where

e_{i}=1

. Each broad set

S_{j}

fixes

(e_{1},e_{2},e_{3})

. Points: 300 start states of each narrow set

S^{\varepsilon}_{j}

Method	Cost	$E_{\log}$ (nat) $\downarrow$	Diagnostic
Foresight-SMC, twist:
tempered classifier	5.24	$0.13\pm 0.03$	77 % in $B$
classifier	5.24	$0.18\pm 0.01$	18 % in $B$
$e^{g_{\theta}}$	5.24	$0.47$	16 % in $B$
Foresight-SMC, $L=1$	5.24	$1.26$	ESS 4–15
Reference-proposal SMC	5.24	$1.03$	46/64 zero

Appendix

Table F.1 : Alanine tripeptide: eight broad queries with diagnostics. Cost:

10^{8}

evaluations for all eight queries, excluding one-off costs (Appendix F.2 ). Diagnostic: fraction of particles in

B

T

(SMC), effective sample size out of

4096

(

L=1

), islands with a zero estimate out of

64

, or RSE of a single run.

\pm

: SE over three training seeds.

Log error $\lvert\log\hat{p}-\log p\rvert$ (nat)
	$S^{\varepsilon}_{1}$	$S^{\varepsilon}_{6}$	$S^{\varepsilon}_{4}$	$S^{\varepsilon}_{8}$
$10^{3}\,r_{S}$	$0.55$	$4.0$	$4.6$	$16$
Foresight-SMC	0.17	0.17	0.21	0.04
Foresight-SMC, untempered classifier	0.02	0.31	0.12	0.17
Foresight-SMC, twist $e^{g_{\theta}}$	0.38	0.43	0.52	0.03
Foresight-SMC, $L=1$	2.66	0.22	1.14	0.09

Appendix

Table F.2 : Alanine tripeptide: four narrow start sets, ordered by increasing

r_{S}

, at about

6.6\times 10^{7}

evaluations per query (learned-RC AMS: two to three times more). WNV ratio: relative to Foresight-SMC with the tempered classifier twist, the default variant; above one means more work for the same precision. Reverse counting provides the reference. Naive Monte Carlo finds no hit. † : effective sample size below

12

, so the WNV is underestimated.

Start	Reference	Tempered	Untempered	Plug-in
$\bar{x}_{1}$	$-20.02$ ( $6.0\,\%$ )	0.14	0.11	0.03
$\bar{x}_{4}$	$-17.72$ ( $5.6\,\%$ )	0.05	0.16	0.07
$\bar{x}_{6}$	$-18.03$ ( $6.6\,\%$ )	0.11	0.23	0.14
$\bar{x}_{8}$	$-16.42$ ( $5.3\,\%$ )	0.18	0.11	0.09

Appendix

Table F.3 : Alanine tripeptide, point starts at the narrow-set centers: reference

\log p

(RSE) from an independent Foresight-SMC run (untempered twist,

64

islands) and absolute log errors of Foresight-SMC (

8

islands) with the two classifier twists and of the classifier plug-in.

	Residue 1	Residue 2	Residue 3
$\mu_{B}$	0.26	0.25	0.23
Prediction, $S_{1}$	0.40	0.43	0.38
Foresight-SMC, $S_{1}$	0.40–0.53	0.32–0.54	0.38–0.53
Prediction, $S_{8}$	0.09	0.10	0.09
Foresight-SMC, $S_{8}$	0.10–0.13	0.08–0.12	0.09–0.16

Appendix

Table F.4 : Alanine tripeptide: fraction of end states in the upper basin of

B

(

\psi_{i}\in[-30^{\circ},150^{\circ})

) for residues

1,2,3

. Prediction: initial states

Y_{0}\sim\mu_{B}

of relaxations, weighted by

\mathbf{1}_{A\cap S}(Y_{T})

. Foresight-SMC: range over three training seeds.

Figure F.2 : Alanine tripeptide: paths of Foresight-SMC over $T=16000$ steps from the broad set $S_{1}$ (top) and the narrow set $S^{\varepsilon}_{1}$ (bottom), on the $(\phi_{i},\psi_{i})$ plane of each residue (columns) over the free energy of Figure F.1 . Each row shows 20 paths from a start state (white) to an end state in $B$ (magenta), chosen so that the end states follow the full sample.

Figure F.3 : Alanine tripeptide: potential energy $\beta_{\mathrm{th}}^{-1}U(x_{t})$ , with $\beta_{\mathrm{th}}^{-1}=k_{\mathrm{B}}\times 300$ K, shifted to a minimum of zero, along one path from the top row of Figure F.2 (light: every tenth step; dark: running mean over 201 steps). Above: all-atom structures at the marked steps, with residues in the rare conformation in blue. The residues flip one at a time, and each flip raises the potential. None of 32 reference paths from the same start state flips a single residue within $T$ .

Method	Hits in $B$	$E_{\log}$ (nat)
TPS-DPS	4.4–6.8 %	17.7
Trust-region transport	0 of 32768	$\infty$
Foresight-SMC, $L=1$	14.5–29.4 %	1.26
Foresight-SMC, 250-step blocks, tempered classifier	77 %	0.13

Appendix

Table G.1 : Alanine tripeptide: learned samplers without target-side data. Hits are the fraction of paths that end in

B

;

\infty

: zero estimate.

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Abstract

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