Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling
Authors: Yuanqi Du
Michael Plainer
Rob Brekelmans
Chenru Duan
Frank Noe
Carla P Gomes
Alan Aspuru-Guzik
Kirill Neklyudov
Venue: NeurIPS 2024 spotlight (2024)
Keywords: Transition Path Sampling
Protein Folding
Schrödinger Bridge
Primary area: machine_learning_for_physical_sciences
TL;DR: Lagrangian formulation of Doob's h-transform allowing for an efficient rare event sampling

Abstract: Rare event sampling in dynamical systems is a fundamental problem arising in the natural sciences, which poses significant computational challenges due to an exponentially large space of trajectories. For settings where the dynamical system of interest follows a Brownian motion with known drift, the question of conditioning the process to reach a given endpoint or desired rare event is definitively answered by Doob's $h$-transform. However, the naive estimation of this transform is infeasible, as it requires simulating sufficiently many forward trajectories to estimate rare event probabilities. In this work, we propose a variational formulation of Doob's $h$-transform as an optimization problem over trajectories between a given initial point and the desired ending point. To solve this optimization, we propose a simulation-free training objective with a model parameterization that imposes the desired boundary conditions by design. Our approach significantly reduces the search space over trajectories and avoids expensive trajectory simulation and inefficient importance sampling estimators which are required in existing methods. We demonstrate the ability of our method to find feasible transition paths on real-world molecular simulation and protein folding tasks.

Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling
Yuanqi Du
Michael Plainer* 2.3,4,5
Rob Brekelmans"
Chenru Duan 7,8
Frank Noé 4,9,10
Carla Gomes"
Alán Aspuru-Guzik 6,11
Kirill Neklyudov 12,13
Cornell University
"use School ELIZA
TTechnische Universität Berlin Berlin Institute for the Foundations of Learning and Data
Freie Universität Berlin
Vector Institute Massachusetts Institute of Technology *Depp Principle, Inc. 10 Microsoff Research A14Science 1 University of Toronto Universsit de Montréal
Rice University
Mila Quebec AI Institute
Abstract
Rare event sampling in dynamical systems fundamental problem arising in the natural sciences, which poses significant computational challenges due to an exponentially large space trajectories. For settings where the dynamical system of interest follows a Brownian motion with known drift, the question of condi- tioning the process to reach a given endpoint or desired rare event is definitively answered by Doob's ' hi-transform. However, the naive estimation of this transform is infeasible, as it requires simulating sufficiently many forward trajectories to estimate rare event probabilities. In this work, propose variational formulation Doob's h-trannsfomm as an optimization problem over trajectories between a given initial point and the desired ending point. To solve this optimization, propose a simulation- free training objective with a model parameterization that imposes the desired boundary conditions by design. Our approach significantly reduces the search space over trajectories and avoids expensive trajectory simula- tion and inefficient importance sampling estimators which are required in existing methods. We demonstrate the ability of our method to find feasible transition paths real-world molecular simulation and protein folding tasks.
1 Introduction
Conditioning a stochastic process obey particular endpoint distribution, satisfy desired terminal conditions, or observe a rare event is problem with long history (Schrödinger, 1932; Doob, 1957) and wide-ranging applications from generative modeling (De Bortoli et al., 2021; Chen al., 2021a; Liu et al., 2022, 2023c; Somnath et al., 2023) to molecular simulation (Anderson, 2007; Wu et al., 2022; Plainer et al., 2023; Holdijk et al., 2023), drug discovery (Kirmizialtin et al., 2012, 2015; Dickson, 2018), and materials science (Xi et al., 2013; Selli et al., 2016; Sharma et al., 2016).
Transition Path Sampling, In this work, we take particular interest in the problem of transition path sampling (TPS) in computational chemistry (Dellago et al., 2002; Weinan and Vanden- Eajnden, 2010), which attempts describe how molecules transition between local energy minima metastable states under random fluctuations or the influence external forces. Understanding such transitions has numerous applications for combustion, catalysis, battery, material design, and protein folding (Zeng et al., 2020; Klucznik et al., 2024; Blau et al., 2021; Noé et al., 2009; Escobedo et al., 2009).
"Equal contribution. Correspondence: k. s.neelldo@@maail.comm (Kirill Neklyudov)
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Stochattic Dilievaniaa Equadie (SDE) dice by(Rcc di + S dW,
We spprescimate Afx A.ay with sipa Nolie Mlpx) Training
mite by A) A coey to sumppe by stanlaating SIME
orgnnilia Ba (C(x,0)
This ollows us to samuple trajecteties from the conditioned. SIM
ditteaht sample dueetly
dier (h(tt))) + log.,,on fl) dW,
Figure Given reference dynamics, transition path sampling seeks to capture the conditional or posterior distribution over paths which reach a terminal set ør E B. However, simulating the reference dynamics (blue) can be wasteful since we rarely obtain paths (orange) which reach (the vicinity of) the terminal set B. This is a major challenge for techniques based on importance sampling Monte Carlo estimation, even when adding control term the reference dynamics. By contrast, our approach optimizes tractable variational distribution over transition paths with a parameterization which satisfies the initial and terminal conditions by design.
While the TPS problem is often framed as finding the "most probable path" transitioning between states (Dürr and Bach, 1978: Vanden-Eijnder and Heymann, 2008). we build upon connections between TPS and Doob's 's h-transform (Das et al., 2019, 2021, 2022; Koehl and Orland, 2022; Singh and Limmer, 2023) and seek match the full posterior distribution over conditioned processes.
Doob's h-Trranffrm. For Brownian motion diffusion processes, conditioning is known to be achieved by Doob's h-transform (Doob, 1957; Särkkä and Solin, 2019). However, solving this problem amounts to estimating rare event probabilities or matching a complex target distribution. Approaches which involve simulation of trajectories to construct Monte Carlo expectations or importance sampling estimators (Papaspiliopoulos and Roberts, 2012; Schauer et al., 2017; Yan al., 2022; Holdijk et al., 2023) can be extremely ineffieient if the target event is rare endpoint distribution is difficult to match. Recent methods based on score matching (Heng et al., 2021) or nonlinear Feynman- -acc formula (Chopin et al., 2023) still require simulation during optimization.
Variational Formulation of Doob's h- Transform. In this work, we propose a variational formulation of Doob's h-transform the solution to optimization on the space of paths of probability distributions. We focus on solving for the Doob transform conditioning on a particular terminal point, which is natural in the TPS setting (see Fig. 1). Taking inspiration from recent bridge matching methods (Peluchetti, 2021, 2023; Liu et al., 2022; Lipman et al., 2022; Shi et al., 2023; Liu et al., 2023a), we propose a parameterization with the following attractive features.
Every Sample Matters. In contrast to most existing approaches, our training method is simulation-frree thereby avoiding computationally wasteful simulation methods to estimate rare-event probabilities and inefficient importance or rejection sampling. We thus refer to our approach as being sample-eficiient
2. Optimization over Sampling, We propose an expressive variational family of approxima- tions the conditioned process, which are tractable to sample and can be optimized using neural networks with end-to-end backpropagation.
3. Problem-Informed Parameterization. Our parameterization enforces the boundary condi- tions by design, thereby reducing the search space for optimization and efficiently making use of the conditioning information.
We begin by linking the problem of transition path sampling to the Doob's 's A-transform and recalling background results in Sec. 2. We present our variational formulation in Sec. 3.1 and detail our optimization algorithm throughout Sec. 3.2. We demonstrate the ability of our approach to achieve comparable performance to Markov Chain Monte Carlo (MCMC) methods with notably improved efficiency on synthetic, and real-world molecular simulation tasks in Sec. 5.
2
Background
2.1 Transition Path Sampling
Consider forward or reference stochastic process with states x, and the density of transition probability Pt+at (y)2, = x)) == p(xx+ad = y Iz = x). Starting from an initial point zo = A, the probability density of a discrete-time path is given as
T-
plTT..... Fet 20 A)
II dt
p(ze+dr (x.) p( xo A).
(I)
The problem of rare event sampling aims to condition this reference stochastic process on some event at time T,, for example, that the final state belongs to a particular set æт E B. We are interested in

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sampling from the entire transition path, namely the posterior distribution over intermediate states
p. Fat Fo A, Fr E B) p(aT
B, aT ................................ Iett z0 = A)
(2)
p(T E B zo
Moving to continuous time, we focus on the transition path sampling problem in the case where the reference process is given by Brownian motion. In particular, we are motivated by applications in computational chemistry (Dellago et al., 2002; Weinan and Vanden-Eijnden 2010), where the reference process given by molecular dynamics following either overdamped Langevin dynamies,
dr (V.M))MMVVll
dt + (xM) / 2k dWW,
(3)
or the second-order Langevin dynamies with spatial coordinates He and velocities U,e
dā: Up dt, dūe M (VU)) 788)) di M V2ykkT dW.
(4)
for a potential energy function U, where W. denotes the Wiener process. Note that R is the Boltzman constant times temperature, M is the mass matrix, and y is the friction coefficient.
2.2 Doob's h-transform
Doob's ft-transform addresses the question of conditioning a reference Brownian motion to satisfy a terminal condition such as æт B, thereby providing an avenue to solve the transition path sampling problem described above. Without loss of generality, and to provide a unified treatment of the dynamics in (3), we consider the forward or reference stochastic differential equation (SDE). OoTT
dr. = (xx)) dt + E, dW,,
Jog Po:
(5)
with drift vector field b, N N and diffusion coefficient matrix E, NNN such that G. = 2 _ is positive d 2 We denote the induced path measure as Fore E P(C([O,T] - I (),)), i.e. measure over continuous functions from time to R.N.
Remarkably, Doob's h-transform (Doob, 1957; Särkkä and Solin, 2019, Sec. 7.5) shows that conditioning the reference process (5) on 2T E B results in another Brownian motion process.
Proposition 1. [Jamison (1975, Thm. 2)] Let hy(2z,() == pTT E B xr x) denote the conditional transition probability of the reference process in (5). Then.
P draT (b((r))) + 2G, V, log ho( 1)) dl + E, dW,
To Po
(6)
where Zgr to denote conditional process. The SDE in (6) is associated with the following transition probabilities for < t T,
hg(y, 1)
pc F3 = x. E B)
Pc x, = =x),
hg(r, s)
Note that all of our subsequent results hold for the case when B is a point-mass, with the only change being that the h-flanction becomes a density, hn( pr (B x4 x).
See App. A.1 for proof, and note that (7) is simply application of Bayes rule (1 x IT
pr(T E 21 = pe (y) F3 pr/pr E B] 2) with the unconditioned or reference transition probability as the prior. Furthermore, the conditioned transition probabilities in (7) allow us to directly construct the transition path (2). Using Bayes rule, we have
plT de. Fet z0 A, E B) As(eT de. di) p(7T de .....
y.......al de xo A)
Ag(A,0)
after telescoping cancellation of h-functions and rewriting the denominator in (2) as his A, 0) Thus, we can solve the TPS problem by exactly solving for the h-funetion and simulating the SDE in (6).
Finally, the fi-process and temporal marginals Pr(2/zo A, xT E B) of the conditioned process satisfy the following forward and backward Kolmogorov equations, which will be useful in deriving our variational objectives in the next section. Note, we use (Vz.*) = div(+) for the divergence operator, and we use Prjo.T indicate the dependence on both 2g (via the initial condition of (8a)) and zr E B (via the h- t-transform hg). See App. A. for the proof.
Note that the second-order dynamics in (4) can be represented using
(x(r))
Sr
Gt
00
-M (xt( 70s
0
M
2
3

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Proposition 2. The following PDEs are obeyed by (a) the density of the conditioned process P+jo,T(=) = Pr Ar B) and (b) the h-function hg(x,t) t).
@p)):r(() dt + (1) 2..V, (r, f))) Σ
8²
dz,dx, P(j,r(z)) = (8a)
1hg(a,11) (Vahb(2,t)
0²
(Gt)ss dz, hg(a, 0. Σ
(8b)
δt
ij
Reparameterizing (8b) in terms of sz 1) = log hs(r, t), we can also write
(sp(a,t) t)
0 85(2, 0. (8c)
(V) (Vsg(2, I(Gr)s dr,dz,
dt
3 Method
We first present novel variational objective whose minimum corresponds to the Doob h-transform in Sec. 3.1, and then propose an efficient parameterization to solve for the h-transform in Sec. 3.2.
3.1 Doob's Lagrangian
Consider reference dynamics given in the form of either (3) or (4), with known drift b, or energy U. We will restrict our attention to conditioning on a terminal rare event of reaching given endpoint IT B, along with an initial point zo = A. We approach solving for Doob's 's /--transform via a least action principle where, in the following theorem, define Lagrangian action whose minimization yields the optimal of (x) = Prjo.T (z) and (x) = V log h (z, t) from Prop. 1 and 2.
Theorem 1. The following Lagrangian action functional has a unique solution which matches the Doob h- h-transform in Prop. 2,
S min (" di ʃ de (1 G. (x)or(x)) q,0
(9a)
0q))r)(x) -V 2G, ())r(=))) Σ
O
(Glis dr,On, ()) (9b)
Ot
(0(x) 8(x A), (T(2) 6(z -B.
(9c)
The optimal (x) obeys (8a), and (z) = V loghhB(I = vLsp(I,t) obeys (8b)-(8c).
This objective will form the basis for our computational approach, with proof of Thm. deferred to App. A.2. We proceed briefly to contextualize our variational objective and highlight several optimization challenges which will be solved by our proposed parameterization in Sec. 3.2.
Unconstrained Dual Objective. Introducing Lagrange multipliers to enforce the constraints in (9b)-(9c and eliminating "ro.T, we obtain an alternative, unconstrained version of (9a).
Corollary 1. The Lagrangian objective in Thm. which solves Doob's 's fi-raansfrmm is equivalent to
min max sB(B.T) - =n(A.0) - 1," fax ua.r 4/oo.
@sg + +VVnn, (V),,b))+ (V....)) 0h
qooo satisfies (9c). Note (x) (I t), with s f) log h f) optimality.
This objective is similar to the objectives optimized by Action Matching methods (Neklyudov et al., 2023, 2024). Notably, the objective Cor. is expressed directly in terms of the (log) of the h-function for fixed conditioning information FT B. We also note that the Hamilton Jacobi-style quantity, whose expectation appears in the final term, is zero at optimality in (8c) of Prop. 2.
8² Compared to (8c), we write (G))) da,uz, s8(2, t) (V, G.G.VVssla t)) for simplicity of notation.

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Path Measure Perspective. We next relate our variational objective in Thm. 1 to KL divergence optimization over path measures. Let For denote the law of the reference SDE in (5) with fixed Fo 6000 - A). Let Qü. denote the law of a controlled process similar to (6), but with variational Prjo,T in place of V log hi.
Q Note that the density 9:0.7 of Lörr evolve according to the Fokker- Planck equation in (9b) (Särkkä and Solin, 2019, Sec. 5.2). Using the Girsanov Theorem, the objective in (9a) can then be viewed as a KI. divergence minimization over path measures Qö.r which satisfy the boundary constraints.
dr, = (Ta(r) + 2G, (()o)(())))) di + +E, dW,, 20 = A.
(10)
Corollary 2. The following Schrödinger Bridge (SB) problem
min
DrL. Qörr F.T
(11)
2o.T S.t. Q2 A
24
6B
yields the path measure Fo. associated with the SDE in (6) as its unique minimizing argument. The temporal marginals of OOT are equal to those which optimize the Lagrangian objective in Thm. 1.
Our Lagrangian action minimization thus corresponds to the solution of an SB problem (Schrödinger, 1932; Léonard, 2014) with Dirac delta functions as the endpoint measures. Our objective in (9a) particularly resembles optimal control formulations of SB (Chen al., 2016, 2021 b, Prob. 4.4, 5.3). While it is well-known that the Doob h--rransfrrm (and large deviation theory more generally) plays role in the solution to SB problems (Jamison, 1975; Léonard, 2014), our interest in the transition path sampling problem leads to specific computational decisions below. See Sec. 4 for further discussion.
Challenges of Optimizing (9a). We highlight several distinctive features of our problem which inform the development of new computational methods in Sec. 3.2.
1. First, we perform optimization over the first argument of the KL divergence in (11), indicating that we need to be able to efficiently sample from the conditioned process in (10) or d in (9). This appears challenging due to the nonlinearity of both the reference and variational drifts, b, and U:
2. For given q o,7, can be difficult to solve for U:jo,T which satisfies the Fokker- Planck equation in (9b) or Vs which solves the inner optimization in Cor. 1.
3. Finally, we would like to strictly enforce the boundary constraints on 4t 0.7 or Qü.r to avoid simulating wasting computation on trajectories for which Fr # B.
In fact, our parameterization of 4:oo, in Sec. 3.2 will completely avoid simulation of the SDE in (10) during training (Challenge 1), provide analytic solutions for UrJo.T satisfying (9b) with a given 9:jo.7 (Challenge 2), and exactly enforce the boundary constraints (Challenge 3).
3.2 Computational Approach
We now propose family of Gaussian (mixture) path parameterizations Grjo,T which overcome the computational challenges posed in the previous section, while still maintaining expressivity. We present all aspects of our proposed method the context of the first-order dynamics (3) Sec. 3.2.1, before presenting extensions to mixture paths and the second-order setting (4) in Sec. 3.2.2-3.2.3.
3.2.1 First-Order Dynamies and General Approach
Tractable Drift Utjo, for Variational Doob Objective. We begin by considering modification of the Fokker-Planck constraint in (9b), with all drift terms absorbed into single vector field unjo.
0g o (x) -V (x)) (njor(x)) E(GG)
8²
dn,ōr, (x))
(12)
Øt
2J
For arbitrary 9:jo,T, solving for any (z)) satisfying (12) can be difficult optimization problem, whose solution is not unique without some cost-minimizing assumption (Neklyudov et al., 2023).
To sidestep this optimization, and address Challenge 2, we restrict attention to variational families of 4:jo, E where it is analytically tractable calculate vector field u, 0.7 which satisfies (12).
(9.00)
We first consider the family of Gaussian paths Qo, similar fashion to (conditional) flow matching methods (Lipman et al., 2022; Tong al., 2023; Liu et al., 2023a), with proof in App. B.
5

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Algorithm 1: Training (Single Gaussian)
Algorithm 2: Sampling Trajectories
Input: Reference drift b, diffusion matrix G+
def get_drift(z;, t):
Conditioning endpoints
Evaluate TT 2:(0)
while not converged do
return drift " 40,0,7 (mt) using (13)
Sample f ~ 4(00.7 I')
Sample 24 Pejo.r using (15)
(8)
Sample initial state Io N((A,, 0 min)
return SDESolve(ro- get get_dr 7)
Calculate ", ":,0,7 (xz))
(Iz)) using (13)
(q.0) Calculate " 1ju.r (1z) using (9.0) (Tx). (ma), (14)
Algorithms for training with a single Gaussian path (Alg. 1) and sampling or generating transition paths at test time (Alg. 2). Note that we sample from the marginals fr during training, but gen- erate paths by simulating the SDE (10).
Calculate C 10 ( ) G. 0/0077 (Thm. 1)
Update A + optimizer(f,, VoL) end while return 0
Proposition 3. For the family of endpoint-conditionn marginals 9t)o.T(z)) N( O,T, E.joT)
[1 1 110 - "HO,T" G. "tjo.r"
(9.0) "1u (x)
(x- (ejo.T)
(13)
öt
öt
satissies the Fokker-Planne equation for q:Jo,T and diffusion coefficients G, J
Given (q.0)
"oo, corresponding lt 0,2', we can simply solve for the Veju.T satisfying the Fokker-Planck
equation in (9h) in our variational Doob objective (Thm. 1). Since G. was assumed to be invertible
and the base drift bg is known, we have
1 010 (x) = (G.) (x) b((x)).
(14)
We may now evaluate terms involving Vejo,r in our Lagrangian objective in (9) using (14) directly, without spending effort to solve an inner minimization over U/jo,r' (thus addressing Challenge 2).
Optimization over QrJo,T satisfying Boundary Constraints. Given the ability evaluate (4.0)
"/ for
given 9rjoT E Qg as above, our variational Doob objective in (9a) reduces to single optimization over the marginals qejor of a conditioned process which satisfies the boundary conditions (9c).
We consider parameterizing the mean Pejorr and covariance Σejorr of our Gaussian path 9:jo,7 using a neural network. For simplicity, we consider a diagonal parameterization Σyo.T diag( {o fo,. (..orr.ddd))) 1) We parameterize neural network NNET, [0,T] x R R + R x R which inputs time & and boundary conditions To = A, FT = B, and outputs vectors of mean perturba- tions and per-dimension variances. Finally, using index notation to separate the output, we construct
1 + x-(0)
1|007 E,
where E ~~ &~,,OL
(15a)
(0) p 1
(1
A
++ L 1(1
T 11 (1-4) NN ()D)
(15b)
T
(0)
711
t(1
t
1
t
diag (NNET A, B)D)) T TD-
(15c)
10,TT T
T
Crucially, our Gaussian parameterization addresses Challenge I, that we can easily draw sam- ples 2zj0,7 40o.T from our variational conditioned process (9b) without simulating the corre- sponding SDE with nonlinear drift (10). Further, the coefficients in (15b) and (15c) ensure that, as t + 0 or t + T. parameterization satisfies the (smoothed) boundary conditions by de- sign (Challenge 3). Although we add o² to ensure invertibilty of Σnjo, (see (13)), we preserve qo (2o) = (m A, Ip) Rs 0(zn - A) and (T(TT) = W(IT B, (in R 5(IT - B).
Reparameterization Gradients. Having shown that our parameterization satisfies the constraints (9b)-(9c) by design, we can finally optimize our variational Doob objective with respect to 9.o.T E QG using the reparameterization trick (Kingma and Welling, 2013; Rezende et al., 2014). In particular, for the expectation at each & in (9a), we rewrite
V&E (9)
$tj0, (x) (
(1) (z), G. (9.0) (x)) = EN((DDD)
(9.0)
(9.0) Ve u (glt, e; 0)), G. (g.0) 1/..7 (glt, e; {0))]]
(9.0) where r = g(t, 6; 0) is the mapping in (15) and doooT depends on 0 via /0
(0)
20.T in (13)-(14).

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Full Training Algorithm. In practice, we sample a batch of times |- uniformly from the interval E [0, T". For each time point, we approximate the gradient using pengle-aample estimate of the expectation above (or (9)), which yields a simulation-free training procedure. The full training algorithm is outlined in Alg. 1.
Sampling of Trajectories. While we sample directly from 9.oo. during training, we can sample full trajectories which obey this sequence of marginals at test time (Alg. 2). In particular, we simulate SDE trajectories with drift u (4 (x) and diffusion coefficient G, using appropriate solver. Note that this generation scheme sidesteps computationally expensive evaluation of the force field base drift (z+). We visualize example sampling trajectories in Fig. 2.
(0)
3.2.2 Second-Order Dynamics
To handle the case of the second-order dynamics in (4), we can adapt our recipe from the previous section with minimal modifications by extending the state space z E RD include velocities 0, with (I, δ) E R²D_ However, note that the dynamics in (4) are no longer stochastic in the spatial coordinates F. To ensure invertibility of G, and existence of the h-trransfomm we add a small nonzero diffusion coefficient in the coordinate space I, so that the reference process in Eq. (5) is given by
Et Dr
8r (xx))) = -M -V- 708
SminID
0
E
M
1/8
V2kkBT
(16)
All steps in our algorithm proceed in similar fashion to Sec. 3.2.1. We now parameterize qejo, (æ, 8) using NNET, [0, x R2DD x R2D + IR20 x R2DD, which outputs mean perturbations and per- dimension variances to calculate 4 " 1fjor and Σ 2 Σ 2 and sample (z, 0), as in (15). Note that we parameterize Σ i +/o.T separately, matching the block diagonal form of (16). We
calculate (,(9) / (z, 8) = 1oo, rjo.r from (2, [u 1uj,,2" "coo as in (13)-(14), with
a(0) f(q)
(B(a)
G. (+ given by (16). The Lagrangian objective in (9) minimizes the norm of the (g)
concatenated vector "jj, (2, J), which depends on the reference drift 6,(T, D) in (16).
3.2.3 Gaussian Mixture Paths
Note that the true Doob li-tranffrrm may not yield marginal distributions which are unimodal Gaussians as in the previous section. To increase the expressivity of our variational family of conditioned processes, we now extend our parameterization to mixtures of Gaussians, 9:Jo,T E Q MaG- 1K Given a set of K mixture weights w == {w*] Jkm1 and component Gaussian paths {4/j, km the
following identity allows us to obtain the drift (g:0)
"oo of the corresponding mixture distribution quju,T-
Proposition 4. Given a set of processes (2) and mixtures weights w², the vector field satisfying the Pokker-Plancee equation in for the mixture (1)))((x)) = Ex w (/jor(()))) is given by
w (x)
ejo (x)
(q.0)
E wi (()
u
"#JO,T
(g.k)
(x),
(17)
(9.6) where ug (x)) satisfies the Fokker-Planck equation in (12) for (x). This identity holds for both first order dynamics in spatial coordinates only or recond-orded dynamics in x = (z, 8).
Finally, we calculate (q.9)
"210.7 (z:) by comparing (9.0) ". (x:) for the mixture of Gaussian path 2rjo.
@Ko to the reference drift (z) as in (14), and proceed to minimize its norm as in (9). In practice. we use Gumbel softmax reparamerization gradients (Maddison et al., 2016: Jang et al., 2017) to optimize the mixture weights {w* ] Tk=i alongside the neural network parameters {
]km for each "eo.T 1k= and either first- or second-order dynamics.
Gaussian component (u10,,, (0) 1K
Related Work
(Aligned) Schrödinger Bridge Matching Methods. Many existing 'bridge matching" approaches (Shi et al., 2023; Peluchetti, 2021, 2023; Liu et al., 2022; Lipman et al., 2022; Liu et al., 2023b) for SB and generative modeling rely on convenient properties of Brownian bridges and would require calculating h-transforms simulate bridges for general reference processes. Our conditional
7

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Gaussian path parameterization is similar to Liu et al. (2023a); Neklyudov et al. (2024), where analytic bridges are not available for SB problems with nonlinear reference drift or general costs.
Somnath et al. (2023); Liu et al. (2023b) attempt to solve the SB problem given access to aligned data Jo, 2T 96rr assumed to be drawn from an optimal coupling. While the method in Somnath et al. (2023) involves approximating to simulate a Markov process. However, De Bortoli et al. (2023) use Doob's ' / f1-transform to argue the learned Markov process will not preserve the empirical coupling unless gour is the optimal coupling for the SB problem, and show that 'augmented" vo,t which conditions on Zo can correct this issue.
h-transform, their goal is obtain unconditioned vector field e
After training on a dataset of zo, JT ~ data pairs using our method, we could consider using an Qu,7
(augmented) bridge matching objective (Shi et al., 2023; De Bortoli et al., 2023) to distill our learned 0 into a vector field or No, which does not condition on the endpoint. Our use of a Gaussian path parameterization with samples from fixed endpoint coupling and no Markovization step corresponds to a simplified version of the conditional optimal control step in Liu et al. (2023a).
Transition Path Sampling. We refer to the surveys of Dellago et al. (2002); Weinan and Vanden- Eijnden (2010); Bolhuis and Swenson (2021) for an overview of the TPS problem. Least action (Freidlin and Wentzell,
principles for TPS have long history, building upon the Freidlin- 1998) and Onsager-Machlup (Onsager and Machlup, 1953; Dürr and Bach, 1978) Lagrangian func- tionals in the zero-noise limit and finite-noise cases. In particular, the Onsager-Machlup functional relates maximum a posteriori estimators or 'most probable (conditioned) paths' to the minimizers of action functional similar to Thm. 1, where example algorithms include (Vanden- (Vanden-Eijnden and Heymann, 2008; Sheppard et al., 2008). By contrast, our approach targets the entire posterior over transition paths using expressive variational family. While Lu et al. (2017) provide amalysis for the Gaussian family, we draw connections with Doob's h-transform and extend to mixtures of Gaussians.
Shooting methods are among the most popular for sampling the posterior of transition paths. From a path that satisfies the boundary conditions (obtained, e.g., using high-temperature simulations), shooting picks points and directions to propose alterations, then simulates new trajectories and accepts or rejects using Cetropolis-Hastingg (MH) (Juraszek and Bolhuis, 2008; Borrero and Dellago, 2016; Jung et al., 2017; Falkner et al., 2023; Jung al., 2023). While the MCMC corrections yield theoretical guarantees, shooting methods involve expensive molecular dynamics (MD) simulations and need to balance high rejection rates with large changes in trajectories. One-way shooting methods sample paths efficiently but yield highly correlated samples. Two-way shooting methods, which we compare to in Sec. 5, more expensive but typically sample diverse paths faster. Recent machine learning approaches (e.g. Plainer et al. (2023); Lelievre et al. (2023)) aim to reduce the need for MD.
Finally, various related methods rely on iterative simulation of SDE in (10) during training to learn the control drift term. Yan et al. (2022); Holdijk et al. (2023) are motivated from the perspective of stochastic optimal control, while Das et al. (2021); Rose et al. (2021) develop actor-critic methods using closely-related ideas from soft reinforcement learning. The variational method in Das et al. (2019) optimizes the rate function quantifying the probability of the events, while Singh and Limmer (2023) solves the Kolmogorov backward equation to learn the Doob's h- h-traanffrmm However, all of these methods may be inefficient if the desired terminal state is sampled infrequently.
Experiments
We investigate the capabilities of our approach across variety of different settings. We first illustrate features of our method on toy potentials before continuing to real-world molecular systems, including a commonly-used benchmark system, alanine dipeptide, and a small protein, Chignolin. The code behind our method is available at https:/ // /ttbttttt.bb../. Before diving into the experiments, we introduce the evaluation procedure and baseline methods.
Evaluation metries. In our evaluation, we emphasize two key quantities: accuracy and efficiency. Efficiency is evaluated by the number of calls to the potential energy function, which requires extensive computation and dominates the runtime of larger molecules. For accuracy, we evaluate the log-lik likelihood of each sampled path and the maximum energy point (saddle point/transition state) along each sampled path. A good method samples many probable paths (i.e., high log- likeliho)ooo) and an accurate transition state (i.e., small maximum energy). See App. D for further details.
Baselines. We compare our approach against the MCMC-based two-way shooting method with uniform point selection with variable or fixed length trajectories. We found that two-way shooting
8

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(a) MCMC
(b) Ours
(a) Single Gaussian
(b) Mixture of Gaussians
Figure 2: Comparing path histograms and trajectories of TPS using fixed-length two-way shooting and com- paring with our variational approach.
Figure 3: Illustration of the expressivity of unimodal Gaussian versus mixture of Gaussian paths sym- metric potential with two transition path modes.
Method
Evaluations
Max Energy
MinMax Energy
Log-Likelihood (1)
Max Log-Likelihhod (1)
MCMC (variable) 3.53M
-13.77
16.43
-0.75
MCMC
1,03B
-17.80 14.77
-40.21
866.56
17.00
907.15
Ours
1.28M
-14.81 + 13.73
-40.56
858.50 17.61
909.74
Table I: Transition path sampling experiment for Müller- potential. We report the number of potential evaluations needed to sample 1,000 paths, as well as the maximum energy and the likelihood of each path (including mean and standard deviation). The methods marked with 'variable' use variable length setting. MinMax energy reports the lowest maximum energy across all paths (i.e., energy of lowest transition state).
produced the most diverse path ensembles among possible baselines, although the acceptance probability can be relatively low for systems dominated by diffusive dynamics (Brotzakis and Bolhuis, 2016) and might be improved by better shooting point selection. This baseline gives theoretical guarantees about the ensemble and thus can be considered as a proxy for the ground truth. In that sense, our goal is not to beat two-way shooting but approximate it with fewer potential evaluations.
5.1 Synthetic Datasets
Müller-Brown Potential. The Müller- Müller-Brown potential a popular benchmark to study transition path sampling between metastable states. It consists of three local minima, and we aim to sample transition paths connecting state A and state with a circular state definition. Fig. 2, we visualize the potential and the sampled paths and can see that the same ensemble is sampled for both our method and two-way shooting. Our method exhibits slightly reduced variance for unlikely transitions. In Table 1, we can observe that MCMC-based methods require many potential evaluations to achieve a good result, which comes from the low acceptance rate (especially when fixing the lengths of trajectories). Our method requires fewer energy evaluations (1 million vs. I billion) while finding paths with similar energy and likelihood. Note that the likelihood for variable approaches has been omitted, as it is governed by the number of steps in the trajectory and cannot be compared directly.
Gaussian Mixture. We further consider potential in which the states are separated by symmetric high-energy barrier that allows for two distinct reaction channels. In Fig. 3, we observe that single Gaussian path cannot model system with multiple modes of transition paths. Nevertheless, this issue can be resolved using a mixture of Gaussian paths, with slightly increased computational cost.
The Case for Neural Networks. According to our empirical study, the neural network parameteriza- tion of the Gaussian distribution statistics Me jo.r, Eg jo.r is an invaluable part of our framework. As an ablation, we consider parameterizing 14 [o. Eg jo, as piecewise linear splines whose intermediate points are updated using the same gradient- based optimizer as used for neural network training. In App. D.3. we report results comparing the W1 distance of learned marginals using neural network versus spline parameterizations, observing that splines yield inferior results even after an order of magnitude more potential function evaluations. We thus conclude that spline parameterizations are not competitive for learning transition paths and continue to focus on our neural-network approach.
5.2 Second-order Dynamics and Molecular Systems
Experiment Setup. We evaluate our methods on real- woal-wrrld high-dimensional molecular systems governed by the second-order dynamics (4): alanine dipeptide and Chignolin. Alanine dipeptide is a well-studied system of 22 atoms (66 total degrees of freedom), where the molecule can be described by two collective variables (CV): the dihedral angles Φ,, v. Chignolin is a larger system consisting of 10 residues with 138 atoms (414 total degrees of freedom) that cannot be summarized as easily. We use an AMBER1 14 force field (Maier et al., 2015) implemented in OpenMM (Eastman et al., 2017) but use DMFF (Wang al., 2023) to backpropagate through the energy evaluations.

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Method
States
Evaluations
Mux Energy (1)
MinMax Energy (J)
MCMC (variable length)
CV
21.02M
740.70 = 695.79
52.37
MCMC
CV
1.29B*
288.46 128.31
60.52
MCMC (variable length)
relaxed
187. 187.54M
412.65 334.70
26.97
MCMC
relaxed
> 10B
N/A
N/A
MCMC (variable length)
exact
> 10B
N/A
N/A
MCMC
exact
> 10B
N/A
N/A
Ours (Cartesian)
exact
38.40M
726.40 + 0.07
726.18
Ours (Cartesian, 2 Mixtures)
exact
51,20M
709.38 162.37
$13.72
Ours (Cartesian, 5 Mixtures)
exact
51,20M
541.26 278.20
247.96
Ours (Internal)
exact
38.40M
-14.62 = 0.02
-14.67
Ours (Intemal, 2 Mixtures)
exact
51.20M
-15.38 = 0.14
-15.54
Ours (Intemal, 5 Mixtures)
exact
51,20M
-15,50 = 0.31
-15.95
Table 2: Transition path sampling for alanine dipeptide. For MCMC methods, compare different state definitions of A, B: 'CV' uses Φ, z angles. 'Exact' uses very small threshold of aligned root-mean-sque -quare deviation (RMSD) around reference states A, B (as in Ours). 'Relaxed' uses a larger threshold of RMSD around A, B. The method marked with * only samples 100 paths due to computational limitations, while others sample 1,000. Fields with N/A are intractable as single trajectory requires more than billion potential evaluations.
CAL
eo I S A
2000
Figure 4: Transition path for the protein Chignolin. The energy plot a transition path in which the protein folds 1, 000 fs, and passes a high energy barrier at 460fs with about 3, 000 kJ/mol.
Alanine Dipeptide. In Table 2, report results for four variants of our models, which either predict Cartesian coordinates or internal coordinates in the form of bond lengths and dihedral angles (compare App. D.4), either with or without Gaussian mixture. For our method, operating internal coordinates yields better results compared to Cartesian coordinates. where the internal coordinates are distributed similarly Gaussians and our network does need not learn equivariances (Du et al., 2022). Similarly, Gaussian mixture paths perform slightly better than single Gaussian path due to the additional expressiveness. We note that paths sampled with Gaussian mixture exhibit larger variance in max energy they represent multiple reaction channels.
We find that prior-informed definitions of the desired initial and target states (i.e., CV) are necessary for MCMC to work efficiently with fixed-length trajectories. Finding these CVs in practice is challenging and only possible in this instance because the molecule small and well-studied. For the larger system size in Table 2. it becomes intractable to use MCMC to connect precise states A, B ('exact') instead of larger regions ('relaxed'), even with single trajectory. Variable length MCMC with relaxed endpoint conditions with CV perform well on this task, but our method is competitive using fewer evaluations and more strict boundary conditions. Fixed-length MCMC, even with
knowledge, can only find 100 trajectories while needing 50 times more potential evaluations compared variable length.
Chignolin. The folding dynamics of Chignolin already pose a challenge and have not yet been well-studied compared alanine dipeptide. We illustrate the qualitative experimental results for this system in Fig. 4. Operating in Cartesian space, our model samples a feasible transition within 25.6M potential energy evaluation calls and a transition with duration of T 1ps.
6 Conclusion, Limitations and Future Work
In this paper, we propose an efficient computational framework for transition path sampling with Brownian dynamics. We formulate the transition path sampling problem by using Doob's h-transform to condition a reference stochastic process, and propose variational formulation for efficient optimization. Specifically, we propose a simulation- free training objective and model parameterization that imposes boundary conditions as hard constraints. We compare our method with MCMC-based baselines and show comparable accuracy with lower computational costs on both synthetic datasets and real-world molecular systems. Our method is currently limited by rigidly defining states A and B to be point mass with Gaussian noise instead of any arbitrary set. Finally, our method might be improved by accommodating variable length paths.
10

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Acknowledgments and Disclosure of Funding
The authors would like to thank Juno Nam and Soojung Yang for spotting the unphysical steric barrier in the original pair of initial and target states, Jungyoon Lee for spotting error in the energy computation, and Guan-Horng Liu, Maurice Weiler, Hannes Stärk and Yanze Wang for helpful discussions. The work of Yuanqi Du and Carla P. Gomes Schmidt Al in Science Postdoctoral Fellowship. a Schmidt Futures program: the National Science Foundation (NSF), the Air Force Office of Scientific Research (AFOSR); the Department of Energy; and the Toyota Research Institute (TRI). The work of Michael Plainer was supported by the Konrad Zuse School of Excellence in Learning and Intelligent Systems (ELIZA) through the DAAD program Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the German Ministry of Education and Research and by Max Planck Society. The work of Kirill Neklyudov was supported by IVADO.
supported by the Eric and Wendy
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A Proofs
A.1 Proofs from Sec. 2.2 (Doob's h-- i-Trrasffrmm Background)
Proposition 1. [Jamison (1975, Thm. 2)] Let lz( = pr(aT E B B x) denote the conditional transition probability of the reference process in (5). Then,
d ((((()))) + V log hoghb(ar 0)) di + E dW,
PoT:
To
(6)
where we use 2 to denote conditional process. The SDE in (6) is associated with the following transition probabilities for < T,
hg(y, 1)
Pe v. IT E B)
pr(y) x.s = x),
hg(r, 8)
Note that of our subsequent results hold for the case when is a point-mass, with the only change being that the h-fumction becomes density, hz(2, pr xt z).
Proof. See Jamison (1975) for a simple proof based on Ito's Lemma, assuming smoothness and strict positivity of h.
Proposition 2. The following PDEs are obeyed by (a) the density of the conditioned process
(z) pr(x z0 A,T B)) and (b) the h-function hg(x,t).
(p))
0²
+ (10)) 2G, V. log (a. 1))) >(G
dn,On, (=),r(z)) =0, (8a)
O1
Öhg(x,t) t) 0l
0²
(z, + Σ (Gt)ss dr, hx(x 0.
(8b)
Reparameterizing (8b) in terms of s3 t) = log hg(s, t), we can also write
Osp(E, b) + (VSg(z, t), G. V.V. 1) + (Vsg(z,t),b + E(Gi)s dr,dz, sB(æ, f) = 0. (8c)
0t
Proof. Let p(pp((ts 24 = x) denote the transition probability of reference diffusion process
0
Φ²
pleet y xr 2) - -V p(z1++ v 21 x)b(+a(v)) + (Gi))
plz++. 1/2, x),
By,
8)
(18)
where (Ge). 1
Now we condition the process on the end-point value DT E B, and we get another kernel, i.e.
p(LT E 1itt = y)
p(z1+== = =yle=ylzz I = IT E B) p(æT E B x, x) p(z = o = x)
(19)
We let (,( p(æT E B x) denote the conditional probability over the desired endpoint condition given H, = æ. According to laws of conditional probability, we can describe how hgg (z, t) changes in time using the unconditioned transition probability
p(ar E B x4 = =))= = dy p(rT B 11tt = 1) p(x+t= = x),
(20)
(2.()
we take the derivative
a on both sides, and get 85
0 fdu p(zeto yl xx) Ohgly, 8) + Op(ezts = -010 x) hely, + (e((ut++)) (21)
Ös
Øs
16

---

Using the FP equation for the transition probability and integrating by parts, we have
0 = f dy p(214+ = y 2 = x)) Ols(y,1 + s) + (Vyhslv,t + +3, + Σ (()) dy.dys
8²
hy +
Note that this holds Vz, henee, we have
Ohgly, 9) + (Vyho(y, t + 8), beto(u)) + Σ
0²
(Gt)ss dy,
hB t + 8) = =0,
0s
without any loss of generality we can set =
@hg(y.s) s)
O²
+ (Vyks(y,s)) 5) bo()) + Σ (Gi)s Oy;
hi (y, 5) = 0.
(22)
0s
as desired to prove the optimality condition in (8b).
To prove (8a), denote p(y, 8) p(a,, y we x) and differentiate p(a, = y 30 =y|30
JT B)
(z.e)) A (xr0) p(y, s) as
0
p(u, = 2 = IT E BB)
0s
1
@hg(y. 8)
8p(y,8)
p(
+ h)(u.s)
=
hs(x,0) 0)
Ös
Ös
1
(V Vyhs(u 8), plu,
ply.s) Σ
Gel dy: holy, s)
0²
hB(2, 0)
8²
-hs(y, 8) V p(y. 8) ())( hB(U,8) 8)
Σ
(G.)ss Ty;Uy, plu: 8)
4
hBly, s)
13(
18(x 0) ply. s)ba(w) p(y, 5)
V 2DV
his (xx, 0)
vyp(y. s). 2DV ho( 5)
0²
h16 0) ply. s)).
h s)
+
(Gi)ss dy,
1z(z 0)
Note that (zz 0) can be pulled outside the differential operator because is a function x. The PDE for the new kernel p(y,s B) = p(u, 1|20 IT E B) (conditioned on the end-point) becomes
i
0s plu, 8 B)- = -VV,, | (b.()) + 2DV, log (o((,,)))) + [(GD) By,
plu, 8 |B).
(23)
which matches the desired PDE in (8a) thereby proving the first two parts of Prop. 2.
Finally, to show (8c), we index time using t in Eq. (22) and change variables hg(r, t) = (=().))
(2.1)
(Vec) b((x)) + > (Gi)ss dz,0z,
(a,,)) = 0.
Ot
(m.)) 0s(r,t) t)
+ (V
+ (V. (V.G.Ve =0
Next, we simplify (V, (V,Gee) = (v.G,e t)) = (Ver(e.), G. Volz t)) (olaa)(V.co G. G.Vs(t,t)) = (t(a.t)
t) + (a (V, t)) to finally write
0s(x,t) (s(x,1)
82
+ (, +
+
Σ (Gt) Dr, s t) = 0
Ot
which demonstrates (8c) since the inner term must be zero
17

---

A.2 Proofs from Sec. 3.1 (Lagrangian Action Minimization for Doob's 's h- h-Transffrm)
We begin by proving Cor. I, whose proof actually contains the initial steps needed to prove our main theorem Thm. 1. In both proofs, we omit conditioning notation qe < 4:J0TT for simplicity and assume qe (z) + O vanishes at the boundary x: + =00, which is used when integrating by parts in a. Corollary 1. The Lagrangian objective in Thm. I which solves Doob's 's h--raasffrm is equivalent to
S = min max (D(B (A( J. de de u/o.
[=gg 81
+ (VS,, + (VS,, bn) + (V.G.Vs))
q/ooT satisfies (9c). Note (x)) Vesa(a,t), with sz t) log (2(2, t) optimality. 4
Proof. Consider the following action functional
ʃ dt f |dx
S = min 9.u
dq((x)
2²
s.t.
= (V d (b)(()( (Gx
+ E(Gi)s
(Gi)ss 0r,Or; 4e (z),
ØL
ij
go(x) 8(x - A), q (2) 8(x - B).
The Lagrangian of this optimization problem is
2 L ['- dt S dr qe(ue, + 8t
Oy: (v, 2G at
0 qt
E ((e) dr,On,
where 8t is the dual variable and we omit the optimization arguments, with S = ming v max, C. Swapping the order of optimizations under strong duality, we take the variation with respect et in an arbitrary direction h2. Using G. = GT,, we obtain
SL [b.] (c((G, G.),,, Ju,, ba) 22 du,
122 = Vst.
(24)
Substituting into the above, have
L [/ dt f /de
du
VoL G.Vos) + + (V.bb)) si v.VV
8t
(25)
at
Integrating by parts in and in assuming that qt (:) (x) + 0 + to0, yields
f da ạT
f da 9080
+
f-"
dx
q: Os:
q Ve, C. - 91 V5t 01 + (es,, CVV.)
[ de grst
f de goso
T
Ust 9 Vs. G.Vss. 44 (vss, -V(V
+
dt
dx
Ot
=f dx STST dx 9080
(8s
+ (Vst, (.ss.).vs. G.V::) + (V + (V.G.Vs.)
dx qe
Ot
(26)
where in the second line, we integrate by parts x again. Enforcing 9T (x) 5(x B) and (x)) 5(x A) and recalling S = min, max, C after eliminating Ut, we recover the optimization in the statement of the corollary.
Theorem. 1. The following Lagrangian action functional has unique solution which matches the Doob h-transform in Prop. 2,
S min [' dt { dx (1), G2 vz),)((z))
fdx
(27a)
2.0
(z)) = -V + 2G, (x)))()))) + E(G.), Øt
8²
s.t.
dx,0z ())
(27b)
ij
(u(x)= = 8( - A),
(τ( = 8(ι - B).
(27c)
'Compared (8c), write E.g (G.)
0 ( s5 t) (V, V. t)) for simplicity of notation.
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---

Namely, the optimal (z)) obeys (8a) and the optimal (z) = Va log (x) = (2s(2, t)
follows (8b) or (8c).
Proof. The proof proceeds from (25) above,
min max min max So dt [ dæ 8 (Vss, + s((V. - V VV)) (28)
4 We first show that the optimality condition with respect to 8, yields the Fokker- Planckk equation for dr
in Prop. 2 (8a), before deriving the PDE in (8b) the optimality condition with respect to q:-
Optimality Condition for (27) recovers Prop. (8a): The variation with respect to st of (28) is simple, apart from the intermediate term. For a perturbation direction he, seek
/ dr
8(-)
d [ dx ge (V(s + Eha), G. (sv 88t h: de
e=
where indicates the functional on the right hand side. Proceeding to differentiate with respect to e. use linearity to pull d inside the integral and apply it first to obtain (st + the) h,- Using the product rule, recognizing the symmetry of terms, and evaluating at & = 0, we are left with
de
Sst be ---1 [-2 dr qe (be.. G.V.s) @ [ dx be (2 (2(V...GVV))) / dx 8(-)
(29)
where in (i) integrate by parts
We are now ready to set the variation of (28) with respect to 8, (in an arbitrary direction h2) equal to zero. Using (29), we have
G [br] = 0
0q.
+ 9.G.V.)
+ (V. (V,
8s:
Ôt
0q. (, q1 b, + + (V,
(30)
Øl
which matches the desired optimality condition for the conditioned process in Prop. 2 (8a).
Optimality Condition for (27) recovers Prop. 2 (8b); Starting again from (28), we take the variation with respect to qt- First, we repeat identical steps (integrate by parts in both x and t) to reach (26),
== f da 9T&T
f goso - S/" di f
Ös: 0t
q:
(Vol, G. + (V sv, by) GVs))))
where it is now clear that taking the variation with respect to qu and setting equal to zero yields
8C
Ds:
bh] = 0
+ (Vot, G.Vsi) + (Vsr, b,) + (V, G.
(31)
ōge
a
which is the desired PDE for s(x, () log hus (x, 1) in (8c). To obtain (8b), we note identity used simplify the last term
8²
log he (V, (v, 1 G.vh: - (Ve + he
Σ
VV. ( v.v.).
dn,dr,
Now, substituting s(x, f) = log 1o((,, t) into Eq. (31) and abbreviating log hs(,,t) = loog /e (+), obtain
1 ah, hh Ol
n (Vhe, G.Vhe) + (Vhe, bb)
1 hee
1
The, G.Vh.) + Tr (v.G. = 0,
+
(h,(x)
+ (V(())).bll)) G.V.)) = 0, (32)
Øt
which matches (8b) as desired.
The last equation defines the backward Kolmogorov equation for the diffusion process with the drift ba()) and covariance matrix G2, i.e. the function he(z) defines the conditional density h2 = p(aT E B' xr = .r) for some set B', which agrees with the forward process with the same drift and
covariance. The boundary condition gr (x: 6(x B) together with the backward equation define the unique solution to this PDE. Since the PDEs and the boundary conditions are the same as in
Doob's 's h hi-rrassfomm we have he hz(z) p(øT B z).
19

---

Corollary 2. The following SB problem
S min s.l. Q5
DEL Q6.r
(11)
yields the path measure o.r associated with the SDE in (6) its unique minimizing argument. The temporal marginals of o.r are equal to those which optimize the Lagrangian objective in Thm.
Proof. We use the Girsanov theorem (Särkkä and Solin, 2019, Sec. 7.3) to calculate the KL divergence between the following two Brownian diffusions with fixed initial condition 2y = A,
dr, = by(mz) dt + E, dW,, dr, = (bx(z)) + 2G, (H0. dt + E dW,, In particular, noting the difference of drifts is (z() + 2G, (0)))(())) b((2z) 2G v1)o,((1)). the likelihood ratio is given by
For
(33)
(34)
d (To. exp UPre O.T
1 /
T
(G.) 2Gr vejo.r (xr))
(35)
p(30, .TT
- -/ 2(, (x))) Gr
We finally calculate the KL divergence, noting that, after taking the log, the expectation of the integral [(),W, in the final term vanishes,
2 S." de I sdeee (x)))()))) (Ugoor Ge (x)))
DRI(Qut PoT -2
(36)
which matches (9a) up to a constant factor of does not change the optimum. We finally compare to the constraints in Thm. 1. First, it clear that the diffusion in (34) satisfies the Fokker-Planck equation (9b) (Särkkä and Solin, 2019, Sec. 5.2). We respect (9c) by optimizing over endpoint-consttraiee path measures, which yields
S min Q st. Q=
DR DRI1Qur Tpoe
(37)
as desired.
B Gaussian Path Parameterizations
Proposition 3. For the family of endpoint-condditonen conditionedd marginals (+)a,((x) N(x 0, EcjoTT)
(9.0) H/u (x)
Hptjo.T
Øt
+
[ 0E -=1
2 Øt
U/o.T
Ge Soju.r (x Hejo.r)
(13)
satisfies the Pokker-Planck equation (12) for qejo, and diffusion coefficients G2 E.
Proof. Consider the following identities for the Gaussian family of marginals (xx N(zlu Et).
where we omit conditioning gr +- 48jo, for simplicity of notation,
1
d
log (z(z) = - - (x ' (x me) log(2T))
log det Σ,,
(38a)
2
2
V, log (z) 1 - -wz)
(38b)
i
log qe (x) (xc "=)"E²" "ー- 0μ:
Mi' E (x
1
θΣ,
at + ((x ((1=-))*
m4)
tr
Dr
at
01.
2
Øt
(38c)
We begin by solving for a vector field (z) that satisfies the continuity equation (where uf denotes the drift of ODE)
dy:
- (V., q -1 (.., ug) + (V V log 9: - -(V 2i) - / x logge, qt. ug))
0t
(39)
Ot
20

---

The vector field satisfying this equation is
Opp dl
2- mce)
(x(x)
+
(40)
which we can confirm using the identities in (38). Indeed, for the terms on the RHS of Eq. (39),
00.
(Vas uf) - 2 Ex
0t
Ope.
022 = (a pc) Ōt
VV log qt uf)
E, (x pe):
E,
Ōt
Putting these terms and the time derivative from (38c) into Eq. (39) conclude the proof.
However, we are eventually interested in finding the formula for the drift u, that satisfies the Fokker- Planck equation in (12). That is, to describe the same evolution of density @g(a) the relationship between Ue and ug is follows
8t
(x) Øt
-Vas a. qe(e)) (Vas lt Un + V Gegt Va log
(Var lt we Vvz G.V.n.))
Vxi qt (wt - G. V. log
Finally, we use the identities in (38) to obtain
0μe
وc D; (x m4) - G.I (x - pe)
u ui + G.V. log q = Ot
+
2 Ot
[1 0., y-1 - G.2i (x-Hi) Hi)
u1
+
0t
2 dt
Proposition 4. Given set of processes (2) and mixtures weights w": the vector field satisfying the Fokker-Plase Planck equation in (12) for the mixture (:jo,((z) = Ex w (x) is given by
* (x) Σラ= wi (()
(q.0) (x) H0.
u (g.k) (x), (x),
110,7
(17)
where (9.8) (x) satisfies the Fokker-Plance equation in (12) for (x). This identity holds for f first-order dynamics in spatial coordinates only or second-orde dynamics in x = (ø, 8).
both
Proof. See Peluchetti (2023) Theorem and its proof in their App. A.
C Extended Related Work
C.1 Machine Learning for Molecular Simulation
The main dilemma of molecular dynamics comes from the accuracy and efficiency trade-off- accurate simulation requires solving the Schrödinger equation which is computationally intractable for large systems, while efficient simulation relies on empirical force fields which is inaccurate. Recently, there has been a surge of work in applying machine learning approaches to accelerate molecular simulation. One successful paradigm is machine learning force field (MLFF) which leverages the transferability and efficiency of machine learning methods to fit force/energy prediction models on quantum mechanical data and transfer across different atomic systems Smith et al. (2017); Wang et al. (2018). More recently, increasing attention has been focused on building atomic foundation models to encompass all types of molecular structures Batatia et (2023); Shoghi et al. (2023); Zhang et al. (2022).
21

---

However, the maximum energy does not account for the fact that the transition path needs to be sequential, and each step needs to be coherent based on the previous position and momentum. For Sampling is a classical problem in molecular dynamics to draw samples from the Boltzmann distribu- tion of molecular systems. Classical methods mainly rely on Markov chain Monte Carlo (MCMC) or MD which requires long mixing time for multimodal distributions with high energy barriers Rotskoff (2024). Generative models in machine learning demonstrate promises in alleviating this problem by learning to draw independent samples from the Boltzmann distribution of molecular systems (known as Boltzmann generator) Noé et al. (2019). Numerous methods have been developed to utilize generative models as proposal distribution for escaping local minima in running MCMC methods Gabrié et al. (2022). However, one critical issue is that generative models rely on training from samples. Although recent advances have been developed to learn from unnormalized density (i.e.. energy) function, the training inefficiency limits their applicability to solve high-diment high-dimensional molecular dynamics problems. To circumvent the curse of dimensionality for the sampling problem, another branch of work study to learn coarse-grained representation with neural networks Sidky et al. (2020). For broader literature of applying machine learning to enhanced sampling, we refer the reader to Mehdi et al. (2024).
D Further Experimental Details
D.1 Evaluation Metries
To assess the quality of our approach in terms of performance and physicalness of paths, we compare them under different metrics to well-established TPS techniques. One important describing factor of a trajectory is the molecule's highest energy during the transition. These high-energy states are often referred as transition states and less likely to occur but they determine importance factors during chemical reaction such as reaction rate. As such, we will look at the maximum energy along the transition path and use to compare the ensemble of trajectories more efficiently. The main goal is to show that lower energy of the transition states be sampled by the methods.
this, we also compare the likelihood of the paths (i.e., unnormalized density) by computing the probably of being in the start state p(2o) and multiplying it with the step probability such that
N-2 JI...,,, - p(zo)) II
x (2++1 /x:).
(41)
For the step probability ग, solve the Langevin leap-frog implementation as implemented in OpenMM to solve (( x; + dt be (z)), dto?). As for the starting probability, we compute the unnormalized density of the Boltzmann distribution for our start state and assume that the velocity v can be sampled independently (Castellan, 1983, Sec. 4.6)
U(z) (U)())
p(z, u) x exp (xmp(Tbr KgT
(42)
with the Boltzmann constant kB and the diagonal matrix M containing the mass of each atom.
As for the performance, the number of energy evaluations will be the main determining factor of the runtime for larger molecular systems, especially for proteins. We hence compare the use of the number of energy computations as proxy for tests, this number aligned with the relative runtime of these approaches.
relative measurements. In our
D.2 Toy Potentials
The toy systems move according the following integration scheme (first-order Buler)
st+1 = xt - (x.(x.) + vdl diag(E) E, E e~ 10,,1)
(43)
following the definition of our stochastic system in Sec. 2.2 with a time-independent Wiener process, where& & is a constant iime-indeeendent standard deviation for all dimensions.
22

---

Müller-Brown. The underlying Stillee-Browe potential that has been used for our experiments can be written
U2x v) = 200 exp( (- - 10y²)
100 exp(- (-p - 10. (v -0.)))
(44)
170 exp (-pp (0.5 + x)² + +11 (Ir + 0.5) (y 1.5) - 6.5 6.5.60.... (y 1.5)9)
15 exp (1+ 0.6 (x 1) (y 1) 1)
0.7 (y 1)
We used first-order Euler integration scheme to simulate transition paths with 275 steps and dt of 10 10-*s. & was chosen to he 5 and 1, 1,000 transition paths were simulated. We have used an MI.P with four layers and a hidden dimension of 128 each, with swish activations. It has been trained for 2,500 steps with a batch size of 512 and a single Gaussian.
In Fig. 5a, we compare the likelihood of the sampled paths. We can see that one-way shooting takes time until the path is decorrelated from the initial trajectory. which is shorter and thus has a higher likelihood. All MCMC methods exhibit this behavior, which is typically alleviated by using warmup period in which all paths are discarded. After that, all methods exhibit similar likelihood, with our method having slightly lower likelihood. Looking at the transition state (i.e., maximum energy on the trajectory) in Fig. 5b reveals that all methods have a similar quality of paths.
12122
(a) Log Path Likelihood
(b) Maximum Energy
Figure 5: In we compare the log likelihood of sampled trajectories, where higher likelihond is generally more favorable. The plot h shows the maximum energy of each individual trajectory. A high maximum energy means that the molecule needs to an excited state during the transition, making it less likely to occur under lower temperatures.
We can further analyze the quality of our method by investigating the difference between the "ground truth" marginal (x) and the learned marginal (z). For this, we compute the Wasserstein WI distance (Flamary al., 2021) between the marginal observed by fixed-length two-way shooting (which we assume to be close to the ground truth) and our variational approach. We observe a mean WI I distance of 0.130 + 0.026 and visualize along the time coordinate (in steps) in Fig. 6.
6:30
Figure 6: In this figure, we compare the Wasserstein WI distance between the marginals. The densities are almost identical at the beginning and end state. Note that at the third local minimum of the Müller-Brown potential along the trajectory (i.c., reached at about 200 steps), the marginals align more closely as well.
Dual-Channel Double-Well To demonstrate the advantage of mixtures, we have used the two- dimensional potential
U(x v) + 2 exp(( + 12y²))
1-eepp(-(11) (a + 0.5) 12y²))
(45)
- 11-exp(-(112) (x - 0.5)² 2 12yy)) + x + 3
23

---

In this case, we have used dt = 5x 10 with a transition time of T = 1s and & = 0.1. As for the MLP, we have used the same structure in the Müller-Brown example but trained it for 20,000 iterations. The corresponding weights to Prop. 4 are [ 2] and are fixed for this experiment and hence w Ø 0.
D.3
Neural Network Ablation Study
"tJo,T impact the quality of trajectories on the Müller-Brown potential. For this, we compare linear and cubic splines (with 20 knots) with neural networks. As metric to estimate the quality, compare the Wasserstein W1 distance between the learned marginal (z) and the marginal observed by fixed-length two-way shooting (i.e., baseline). We notice that using linear splines results in the highest W1 distance, while cubic splines improve the quality. Using neural networks, however, yields the best approximation.
In Fig. 7 we compare how different parameterizations of "" and x1
(0)
We have fixed the computational budget for all systems, which means that have trained splines for more epochs than the neural network (since they are slower to train). For high- high-dimensional systems, the runtime is mostly determined by the number of potential evaluations and not the complexity of the architecture. We thus conclude that the additional expressivity provided by neural networks is necessary for more complicated (molecular) systems and does not introduce much computational overhead.
Calic Sarne
Figure 7: We compare the Wasserstein WI distance between the marginals with different parameterization techniques of n p and (+)
(8)
"
D.4 Molecular Systems
To simulate molecular dynamics, we rely on the AMBER14 force field Suaber14/proteinff144S5 Maier et al. (2015)) without a solvent, as implemented in OpenMM (Eastman al., 2017). As OpenMM does not support nuto-differentiation, we do not use OpenMM for the simulations themselves, but utilize DMFF (Wang et al., 2023) which is a differentiable framework implemented in JAX (Bradbury et al., 2018) for molecular simulation. This is needed (0) because during training we compute VoU F2joor N(u ( = where the concrete TtJo,T is sampled based on the parameters of the neural network.
For the concrete simulations, ran them with the timestep dt 1fs, with T = 1ps, r 1ps, and Temp = 300K. To compute the MCMC two-way shooting baselines, we use the same settings and consider trajectories failed, if they exceed 2,000 steps without reaching the target.
Neural Network Parameterization. MI.P with ReLUU activation function and 256/511 12 hidden units for alanine dipeptide and Chignolin, respectively. The neural networks are trained using an Adam optimizer with learning rate 10-4.
parameterize our model with neural networks, 5-layer
We represent the molecular system in two ways: (1) in Cartesian coordinates, which are the 3D coordinates of each atoms, and with (2) internal coordinate which instead uses bond length, angle and dihedral angle along the molecule, where we use the same parameterization as in (Noé et al., 2019).
Our state definition includes variance parameter for the initial and target marginal distributions t= and = T, we choose the variance to be 10 which almost does not change the energy of the perturbed system.
Visualization of Transition for Alanine Dipeptide. Fig. 8, we show a transition sampled without any noise from the model with internal coordinates and 2 Gaussian mixtures.

---

Figure 8: Transition path for the alanine dipeptide.
Comparison of Sampled Paths During Training. In our training procedure, the marginal starts with linear interpolation between A and B, which produces very unlikely paths with potentially high energy states. In Fig. 9, we compare how the quality of paths changes depending on the number of training iterations (i.e., the number of potential evaluations). We show the curve for a single Gaussian mixture with Cartesian coordinates. Similar trends can be observed in other settings.
Nusber vinaatt Insurcon
Figure 9: In this figure, we compare the quality of paths based on the current training step (i.e., potential evaluations). We observe that with increasing training time, paths with higher likelihood are sampled.
Loss Curves. In this section, we would like to investigate the training losses of different configura- tions. For this, plot the exponential moving average of the loss (a 0.001) better highlight the trends of the noisy variational loss. Fig. 10 compares the results of different training settings. We can observe that mixtures can decrease the overall loss, but all model variations converge similar loss value.
Culacal
20.00
Figure 10: Visualization of the loss for diffeerntt training setups. These setups are identical to what has been reported in Table 2.
D.5
Computational Resources
All our experiments involving training were conducted a single NVIDIA A100 80GB. The baselines themselves were computed on M3 Pro 12-core CPU.
E
Societal Impact
Our research concerns the efficient sampling of transition paths which are crucial for a variety of tasks in biology, chemistry, materials science and engineering. Our research could potentially benefit research areas from combustion, catalysis, protein design to battery design. Nevertheless, we do not foresee special potential negative impacts to be discussed here.
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NeurIPS Paper Checklist
1. Claims
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Answer: [Yes]
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Reviews:
[{"review_id": "KfDcnjQmTC", "reviewer": "Reviewer_iZfM", "summary": "This paper proposes a variational formulation of the Doob's h-transform to change the problem from the expensive simulations of trajectories to an optimization problem over possible trajectories from given initial point and end point. The model parameterization introduced imposes the desired boundary conditions by design and uses a mixture of Gaussian as the variational family. The variational formulation offers an alternative to expensive simulations and MCMC which is hard to scale. The proposed framework is applied on a simulated case study and real world problem in material science/computational chemistry of transition path sampling  which is a study of molecule transitions betweeb local energy minima and metastable states under random fluctuations. The results are along expected lines where it is possible to scale the solution on computationally intractable problems with MCMC and other sampling based methods.", "questions": "1. I am curious,  similar to Figure 3, the authors can show an experiment with familiy of dtistributions other than Gaussian. How does the expressivity then come to play ? \n2. as for any SDE problem solution, what is the effect of discretization values and parameters in practice as given in line 184-185.\n3. How big is the constraint of fixed length(T) on transition paths in practice as solved in this method ? \n4. What is the behaviour of the problem when there are multiple basins of attraction ?", "limitations": "The authors have addressed the challenges quite well and what the future research directions could be taken.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well written, with sound math, a lot of references on recent work and older literature on the subject. I found the introduction to be a great summary of the work. \n2. The idea is sound and through the use of variational framework to solve a complex problem of estimating rare event probabilties by sampling forward trajectories brings out the desired qualities of sample efficiency, reducing search space and matching a complex target distribution with an easier to sample variational distribution. As also observed in Table1 and 2, MCMC cannot be scaled to certain large scale experiments where Variational inference can give good results. \n3. The work finds application for solving transition path sampling problem which is important in material science and chemistry from what it seems(I am not so well acquanited with those domains.) The case study on protein is well documented and explained. \n4. The figures and illustrations are clean and support the narrative.", "weaknesses": "Minor things\n- Increase font sixe for Table1, bold the important results.\n- I would have liked to see a bit more dicussion on the dimensionality, what are the typical values of D ..", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1: I would have liked to see a bit more discussion on the dimensionality, what are the typical values of D ..\n\nThank you for your valuable feedback. We will discuss this in the next revision of the manuscript. The dimension $D$ depends on the specific system being modeled. For instance, alanine dipeptide with its $22$ atoms results in $D = 22 \\times 3 = 66$. If modeled with second-order dynamics, $D$ doubles to $132$. Chignolin has $166$ atoms and results in $D = 996$ in our experiments. Large molecular systems, such as proteins with explicit solvent environments, can exceed $10,000$ atoms, with some even reaching $100,000$ atoms. As such, TPS approaches must accommodate these high-dimensional systems. \n\n> Q2: I am curious, similar to Figure 3, the authors can show an experiment with familiy of dtistributions other than Gaussian. How does the expressivity then come to play ?\n\nNote that the crucial restriction on the family of marginals is the availability of the analytic form of the vector field satisfying the Fokker-Planck equation in order to avoid the min-max formulation in Corollary 1. We conducted the experiments using Corrolary 1 in toy settings but found it unstable when scaled to real-world systems.\n\n> Q3: as for any SDE problem solution, what is the effect of discretization values and parameters in practice as given in line 184-185.\n\nWe do not discretize time during training, the time variable is drawn uniformly $t \\in [0,T]$ in order to estimate the time-integral. We will clarify this in the final version of the manuscript.\nThese training details can also be seen in the training algorithm uploaded in the rebuttal PDF.\n\n> Q4: How big is the constraint of fixed length(T) on transition paths in practice as solved in this method?\n\nIn general, there are ways to estimate the transition time $T$ [1,2,3], and this problem is much easier than sampling the path itself. However, we consider fixed trajectory length to be one of the limitations of our algorithm and a direction of future work.\n\n> Q5: What is the behaviour of the problem when there are multiple basins of attraction?\n\nOur algorithm is designed to solve the concrete problem statement that trajectories end in a particular state $B$. However, due to the stochastic nature of the dynamics, there are multiple pathways to reach B. The most vivid example is presented in Figure 3, where we consider a symmetric potential, and the sampled paths follow two different ways. Similarly, in the Müller-Brown experiment, despite local minima attracting points, transitions still end in the target state. This illustrates that the algorithm can handle multiple basins of attraction by sampling diverse paths that all converge to the designated end state.\n\n> Q6: Increase font size for Table1, bold the important results.\n\nWe will improve the formatting of the tables and the presentation of our results in the revised manuscript. \n\nReferences\n\n[1] G. H. Taumoefolau and R. B. Best, 2021, “Estimating transition path times and shapes from single-molecule photon trajectories: A simulation analysis,” The Journal of Chemical Physics, vol. 154, no. 11. AIP Publishing. \n\n[2] H. S. Chung, J. M. Louis, and W. A. Eaton, 2009, “Experimental determination of upper bound for transition path times in protein folding from single-molecule photon-by-photon trajectories,” Proceedings of the National Academy of Sciences, vol. 106, no. 29. Proceedings of the National Academy of Sciences, pp. 11837–11844..\n\n[3] E. Suárez, J. L. Adelman, and D. M. Zuckerman, 2016, “Accurate Estimation of Protein Folding and Unfolding Times: Beyond Markov State Models,” Journal of Chemical Theory and Computation, vol. 12, no. 8. American Chemical Society (ACS), pp. 3473–3481.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gHMEQab0vo", "reviewer": "Reviewer_zwM4", "summary": "The submitted manuscript presents a variational formulation of Doob’s h-transform, leading to a novel (simulation-free) computational approach for rare event sampling in transition paths. The task of interest involves conditioning a dynamical system driven by Brownian motion with a known drift term to reach a given endpoint. In theory, this terminal condition can be addressed using Doob’s h-transform, resulting in an associated SDE for the conditional dynamic. However, practical implementation requires knowledge of the h-transform. The authors propose a variational problem formulation that provides the necessary information about the h-transform as its solution. Solving this variational problem results in a computational approach for simulating the conditional dynamic. Compared to existing methods, it is claimed that the proposed approach avoids importance sampling estimators and expensive trajectory simulations. The method is tested on both synthetic and real datasets.", "questions": "- To enhance the practical applicability of the proposed framework, it is crucial to study the impact of inexactly solving the variational problems (9) or (11). Specifically, how do errors propagate to the conditioned dynamical system driven by the SDE (6) when an approximate solution of (9) or (11) is used? Understanding this error propagation is essential for assessing the robustness and reliability of the proposed method in practical scenarios.\n- Is the Optimization problem in (9) well-posed? This means, is the optimal solution unique?\n- In several instances, there are missing commas in the notation for the inner product (e.g., equations (9b) and (12)).\n- Many important mathematical details and assumptions are missing. Most importantly, what are the assumptions on the drift vector field $b_t$ to ensure the well-posedness of the proposed scheme?\n- Regarding Section 3.2.2: Instead of introducing $\\xi_{\\min}$ I suppose that one could also directly work with a pseudoinverse when defining $G_t^{-1}$.\n- In Proposition 4, should it be $u_{t|0,T}^{(k)}$ on the right hand side of the equation for $u$?\n- When referring to probability density functions, it's important to use proper notation. Instead of $\\rho(x_t=x)$, it would be clearer to use $\\rho_t(x)$ depending on the context.\n- In the abstract, you claim that no \"inefficient\" importance sampling estimators are required. However, it would be beneficial to compare this approach with such estimators to demonstrate its advantages more clearly.", "limitations": "- There is no discussion about error propagation arising due to the Gaussian parametrization.\n- The approach is limited to conditioning the sample path on point sets $x_T = B$.\n- Limited statistical evidence is presented in the numerical experiments.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper introduces a novel variational objective to describe Doob’s h-transform, leading to a promising and innovative computational approach for simulating the conditional dynamics of transition paths. Although I appreciate the general approach presented in Section 3.1, I am not convinced by the computational approach proposed in Section 3.2 due to several reasons pointed out below. If these issues can be addressed, I believe that the approach can be further utilized to construct highly efficient sampling strategies.", "weaknesses": "- The proposed approach in Section 3.2.1 introduces certain issues. The connection to Doob’s h-transform established in Theorem 1 is only guaranteed if an exact solution to the optimization problem (9) or (11) is found. However, by introducing the Gaussian parametrization of $q_{t|0,T}$, this guarantee is lost. There is a lack of discussion regarding the implications and potential effects of this parametrization on the overall accuracy and validity of the method.\n- Moreover, the formulation of the task of solving equation (12) given $q_{t|0,T}$ is misleading. In reality, you are solving an inverse problem here. Given the solution of the PDE in (12), your goal is to reconstruct $u_{t|0,T}$, which is generally an ill-posed problem. For instance, introducing uncertainties into the model description via the drift $b_t$ or diffusion $\\Xi_t$ can lead to significant challenges. This scenario is likely when applying the approach in practice under model misspecification. A critical question to address is the robustness of the proposed framework against model misspecification, such as small perturbations in $b_t$.\n- The claim that the approach is simulation-free is not entirely clear. In line 173, it is stated that the Gaussian parametrization allows for the generation of arbitrary samples. However, this assumes that the states $x_{t|0,T}$ are independent for all $t$, which is not true when $x_{t|0,T}$ is a solution of the SDE in (10). The dependence between the states needs to be accounted for in the sampling process, and this aspect seems to be overlooked. \n- I am wondering what the measured quantities in the numerical experiments actually reveal about the correctness of the proposed approach. Evaluating the maximum energy and the likelihood might not provide sufficient information about the accuracy of the estimated distribution. If the sampled paths only follow high likelihood regions, this does not necessarily indicate that the correct distribution has been captured. The objective should be to estimate quantities of interest related to the conditioned SDE in (6). However, it is not clear whether the simulated paths correspond to accurate simulations of (6). For instance, how does the method perform when estimating rare event probabilities or other Monte Carlo estimators with respect to (6)? This aspect needs to be thoroughly addressed to validate the effectiveness of the proposed approach.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">Q1: The connection to Doob’s h-transform established in Theorem 1 is lost due to the Gaussian parametrization of q_t. Lack of discussion regarding the implications of this parametrization on the accuracy and validity of the method.\n\nWe highlight the potential lack of expressivity for the Gaussian parameterization through experiments in Fig. 3 and lines 283-286, where a single Gaussian parameterization fails to capture several modes of the transition path. In this case, our mixture of Gaussian parameterization (proposed in Sec. 3.2.3) is necessary.\n\nThe approach of finding an approximate solution by optimizing within a tractable family is common in variational methods [1]. Indeed, our contributions include deriving a variational objective for Doob’s h-transform in Thm 1. Analysis of the accuracy of particular variational families is difficult to characterize for general problems and is beyond the scope of this work.   \n\n>Q2: The formulation of the task of solving equation (12) given q_t is misleading. Given the solution of the PDE in (12), your goal is to reconstruct u_t, which is generally an ill-posed problem. \n\nAs we point out in line 152, indeed, numerous vector fields $u_{t|0,T}$ can satisfy eq. (12) for the given densities $q_{t|0,T}$. In Proposition 3, our goal is to simultaneously parametrize both $q_{t|0,T}$ and $u_{t|0,T}$ such that they satisfy (9b) (thus avoiding constrained optimization problem).\n\n>Q3: For instance, introducing uncertainties into the model description via the drift b_t or diffusion \\Sigma_t can lead to significant challenges. \n\nWe agree that analyzing the robustness of the proposed approach against model misspecification is crucial for practical applications. However, this analysis is beyond the scope of the current work.\n\n>Q4: The claim that the approach is simulation-free is not entirely clear.\n\nThe sampling of independent $x_{t|0,T}$ for a given time $t$ is justified by parameterizing marginal densities rather than the entire path measure. Indeed, our parametrization in eq. (15) defines the parameters of the marginals that change continuously over time (the SDE that has these marginals can be obtained via Proposition 1). The optimized objective in Theorem 1 relies only on the samples from these marginals, which means that during training, no simulation is needed. When sampling paths, the drift term does not need to be evaluated, allowing for efficient sampling. \n\nWe added a pseudocode of the proposed training and inference algorithm in the rebuttal. We will attempt to clarify this further in the manuscript. \n\n>Q5: Evaluating max energy and the likelihood might not measure the accuracy of the estimated distribution of the conditioned SDE in (6). \n\nIndeed, the likelihood on its own does not capture the distribution of sampled paths. However, transition path sampling is an extremely highdimensional problem (D=66\\*2\\*1000), and thus it is difficult to characterize accuracy of matching the full distribution. We use conventional metrics, such as describing paths by the transition state (point with the highest energy)  [2,3], or estimating the likelihood of trajectories [4].\n\n>Q6: It is crucial to study the impact of inexactly solving the variational problems (9) or (11). \n\nIn this paper, we learn the target path-measure $P^*$, which corresponds to the Doob’s h-transform. Corollary 2 shows that the optimized objective corresponds to the KL-divergence between the parameterization and the reference measure $D_{\\text{KL}}(Q:P^{\\text{ref}})$. Using the Pythagorean relation (see, e.g., Theorem 3.3 in [5]), one can show\n$$D_{\\text{KL}}(Q:P^{\\text{ref}})=D_{\\text{KL}}(Q:P^*)+D_{\\text{KL}}(P^*:P^{\\text{ref}}),$$ \nwhere the last term is a constant. Thus, the minimized objective is the KL-divergence between the parameterization and the target measure. Thank you for your suggestion, we will add the corresponding discussion in the final version.\n\n>Q7: Is the Optimization problem in (9) well-posed? This means, is the optimal solution unique?\n\nAs stated in Theorem 1, the optimization problem in (9) has a unique solution (see proof in Appendix A.2).\n\n>Q8: In several instances, there are missing commas in the notation for the inner product (e.g., equations (9b) and (12)).\n\nBoth equations (9b) and (12) contain the divergence operator rather than an inner product. The notation for the divergence operator is $\\langle\\nabla_x, \\cdot\\rangle = \\text{div}(\\cdot)$, which is introduced in line 98.\n\n>Q9: Missing mathematical details and assumptions. Most importantly, what are the assumptions on the drift vector field b_t to ensure the well-posedness of the proposed scheme?\n\nFor the mathematical details and assumptions, we refer the reader to [6] which defines rigorously the necessary conditions for the Doob’s h-transform and the corresponding PDEs. The necessary assumptions for our result are stated at the beginning of Appendix A.2.\n\n>Q10: Instead of introducing \\xi_{min} I suppose that one could also directly work with a pseudoinverse when defining G_t^-1.\n\nWe will discuss this option in the final version of the paper.\n\n>Q11: In Proposition 4, should it be u_t on the right hand side of the equation for u?\n\nYes, we will correct it.\n\n>Q12: Justification of the “inefficiency” of importance sampling is missing.\n\nFor example, the recent work [7], which relies on importance sampling, requires 120M energy evaluations to output a reasonable transition path.\n\n>Q13: The approach is limited to conditioning the sample path on point sets x_T = B. \n\nIn the transition path sampling, the rare event is usually represented by the point $B$. If several rare events are given, we can run our method several times using different values of $B$. Including conditioning on other sets is a direction of future studies.\n\n>Q14: When referring to probability density functions, it's important to use proper notation. Instead of \\rho(x_t = x), it would be clearer to use \\rho_t(x).\n\nWe will clarify the notation.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jM2hctUDNY", "reviewer": "Reviewer_irHw", "summary": "The Authors of this paper tackle the problem of sampling conditioned SDEs with a specific interest in \"transition path sampling\", i.e. sampling a Langevin-type SDE undergoing a transition between an initial state (or set of states) $A$ and a final target set $B$. Sampling transition paths efficiently can provide a tremendous boost to research in catalysis or drug design. In this paper, the Authors model the transition path distribution with either (i) a parametrized Gaussian process or (ii) a parametrized mixture of Gaussians processes. These families of priors are then optimized by leveraging a variational formulation of the problem developed by the Authors starting from Doob's $h$-transform.", "questions": "Is the $h$-transform related to the committor function in some way? If so, how does your Thm 1 relates to the variational formulation of the committor (see e.g. Eq. 20 of \"Transition Path Theory and Path-Finding Algorithms for the Study of Rare Events\" by Weinan E and Eric Vanden-Eijnden)", "limitations": "Limited experimental evaluation", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The Authors tackle a challenging problem with a novel variational formulation, which is apt to be optimized by leveraging techniques developed by the generative modeling community within ML. A clear strength of the paper is the sound theoretical analysis justifying the proposed method. Interestingly, the choice of Gaussian process (or a mixture of Gaussian processes) variational priors allows the Authors to simplify the algorithm using analytical results and sidestep lengthy calculations at the deployment phase.", "weaknesses": "I find the paper overall clearly written, but I had to go through section 3.2 (Computational Approach) several times to grasp how the method can be deployed in practice. Specifically, I think that it can be improved the explanation of the fact that, because of the Gaussian prior, you only need to model the transition path probability and not the $h$-transform itself, as well as the explanation of the actual optimization step (Reparametrization of Gradients).\n\nThe subscript $0,T$ is introduced for the first time in Eq. (6) without explanation and used throughout the manuscript to label variables related to the conditioned process. It might be useful to clarify this notation explicitly. \n\nThe experimental evaluation is somewhat limited, especially concerning the Chignolin protein. Over the years, many transition path sampling strategies have been developed, as well as very much related enhanced sampling techniques. It would be nice to have a comparison also to different baselines, as well as a discussion on the computational complexity and actual running times of the baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. I find the paper overall clearly written, but I had to go through section 3.2 (Computational Approach) several times to grasp how the method can be deployed in practice. Specifically, I think that it can be improved the explanation of the fact that, because of the Gaussian prior, you only need to model the transition path probability and not the h-transform itself, as well as the explanation of the actual optimization step (Reparametrization of Gradients).\n\nThank you very much for the concrete suggestion. We will improve the presentation to emphasize that we only optimize for $q$ due to the Gaussian parameterization. In the revised manuscript, we also introduce two algorithms (which can be found in the PDF uploaded during the rebuttal) detailing the training and inference procedure. These additions should make our approach more understandable. We welcome any further suggestions you might have on making the paper more accessible. \n\n> Q2. The subscript 0, T is introduced for the first time in Eq. (6) without explanation and used throughout the manuscript to label variables related to the conditioned process. It might be useful to clarify this notation explicitly.\n\nThank you for pointing this out. We have updated the manuscript to explicitly introduce and clarify the subscript notation. We hope this makes the notation clearer and easier to follow throughout the manuscript.\n\n> Q3. The experimental evaluation is somewhat limited, especially concerning the Chignolin protein. Over the years, many transition path sampling strategies have been developed, as well as very much related enhanced sampling techniques. It would be nice to have a comparison also to different baselines, as well as a discussion on the computational complexity and actual running times of the baselines.\n\nThank you for raising your concerns. There is minimal difference among MCMC methods regarding the quality of paths, as they all guarantee convergence to the true distribution. We are unaware of any non-MCMC-based transition path sampling procedures that significantly outperform existing techniques in terms of runtime and trajectory quality.\n\nThus, we view MCMC results not as baselines to surpass but as a gold standard to approximate. Comparing with different variations of MCMC (e.g., improved shooting-point selection [1,2], machine-learned MCMC [3]) might reduce the number of evaluations needed but would not affect the theoretical guarantees. Since the effectiveness of these techniques varies greatly across different systems, we did not include such comparisons. However, we will discuss the computational complexity and running times in the revised manuscript to provide a clearer picture of the performance trade-offs involved.\n\nAs for Chignolin, we showcased we could find plausible paths with a reasonable amount of energy evaluations. However, due to the already high dimensionality of Chignolin it is challenging to sample a meaningful ensemble of transition paths with existing methods. This highlights the advantage of our approach.\n\n\n> Q4. Is the h-transform related to the committor function in some way? If so, how does your Thm 1 relates to the variational formulation of the committor (see e.g. Eq. 20 of \"Transition Path Theory and Path-Finding Algorithms for the Study of Rare Events\" by Weinan E and Eric Vanden-Eijnden)\n\nThe committor functions defined in eq. (18) of [4] is different from the h-transform. Indeed, the committor function $q_{+}$ is time-independent and satisfies eq. (18) in [4], while the PDE for the h-transform (eq. (8b) of our paper) includes the time-derivative. The committor function can be obtained as the integration of $h$ over different event times $T$, however, this is beyond the scope of the current paper.\n\n\nReferences\n\n[1] P. G. Bolhuis, D. Chandler, C. Dellago, and P. L. Geissler, 2002, “TRANSITION PATH SAMPLING: Throwing Ropes Over Rough Mountain Passes, in the Dark” Annual Review of Physical Chemistry, vol. 53, no. 1. Annual Reviews, pp. 291–318.\n\n[2] J. Juraszek and P. G. Bolhuis, 2008, “Rate Constant and Reaction Coordinate of Trp-Cage Folding in Explicit Water,” Biophysical Journal, vol. 95, no. 9. Elsevier BV, pp. 4246–4257..\n\n[3] H. Jung et al., 2023, “Machine-guided path sampling to discover mechanisms of molecular self-organization,” Nature Computational Science, vol. 3, no. 4. Springer Science and Business Media LLC, pp. 334–345.\n\n[4] Vanden-Eijnden, Eric. \"Transition-path theory and path-finding algorithms for the study of rare events.\" Annual review of physical chemistry 61 (2010): 391-420.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ktXRgaVGli", "reviewer": "Reviewer_tKYF", "summary": "This paper proposes a variational formulation of Doob's h-transform, which characterizes the distribution over paths with a given endpoint. Instead of relying on potentially wasteful sampling approaches, the authors propose directly optimizing a tractable variational distribution over transition paths which satisfy the initial and terminal conditions by design. This approach reduces the search space over trajectories and avoids trajectory simulation. Experiments on real-world molecular simulation and protein folding tasks demonstrate the applicability of the approach.", "questions": "- How do you choose the number of mixture components in practice?\n- Why does the MaxEnergy increase when using a mixture distribution as a variational approximation in Table 2?\n- How do you explain the huge variance of the Max Energy of MCMC (variable length) in the first row of Table 2?\n- What are the main limitations of the current approach?", "limitations": "Section 6 has \"Limitations\" in the title, but limitations are not discussed, only future work.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper clearly motivates the problem it tackles, describes the challenges well and nicely introduces the idea behind their method in an illustrative fashion. Overall the paper is very well-written and structured. In particular, it builds up the method piece-by-piece explaining the choices along the way. As far as I can judge the related work seems to be exhaustive. The experiments are done on interesting problems as far as I can tell and I also appreciated that the authors made their code publicly available.", "weaknesses": "My main concern comes from trying to interpret the experimental results and in particular judging the performance compared to MCMC. It seems quite clear from Tables 1 and 2 that the variational approach requires fewer calls to the potential energy function than MCMC, however their performance differences are harder to judge in my opinion. \nMore specifically, in Table 1 the standard deviations are so large that there is basically no meaningful statistical difference between the shown results especially for the Max Energy, but also the Log-likelihood. Now this might mean that MCMC and the presented method both perform well, but its surprising to me that there is that much inherent variation. Similarly in Table 2, the variance for the Max Energy of MCMC in the first line is huge. \nI'm also a bit confused by the Max Energy increases when using a mixture in Table 2.\nTo better understand how the performance of the variational approach improves during training, it would be nice to see a plot that shows the Max Energy as a function of the training epochs.\n\nFinally, I would have liked to see a discussion of the limitations of the approach. Section 6 has Limitations in the title, but does not actually discuss them in any way beyond extensions of the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. To better understand how the performance of the variational approach improves during training, it would be nice to see a plot that shows the Max Energy as a function of the training epochs.\n\nPlease see the supplementary PDF for the plot of max energy as a function of energy evaluations. We will include this plot in the final version of the paper.\n\n> Q2. More specifically, in Table 1 the standard deviations are so large that there is basically no meaningful statistical difference between the shown results especially for the Max Energy, but also the Log-likelihood.\n\nThe variance of the max energy (and log-likelihood) is measured over independently sampled paths from the target path-measure (the Brownian motion conditioned on the endpoint). Note that the large variance is not due to inaccurate measurements or poor performance of the model but due to the diffusion coefficient of the reference measure. Thus, we expect our model to match these numbers requiring fewer energy evaluations rather than surpass the MCMC results in terms of energy or likelihood. Therefore, the performance measure in Table 1 is the number of evaluations.\n\n> Q3. How do you explain the huge variance of the Max Energy of MCMC (variable length) in the first row of Table 2?\n\nIn Table 2, the variance of the max energy is measured across paths of different lengths, which gives very different estimates of the maximum energy. Indeed, when the path’s length is not optimal (too short or too long paths), the maximum energy is very high (the trajectory either has to take shortcuts or can wander into high energy regions). Note that these paths do not affect the estimate of the minimum energy but contribute to the estimate of its mean and variance, which results in the enormous value of the latter.\n\n> Q4. Why does the MaxEnergy increase when using a mixture distribution as a variational approximation in Table 2?\n\nFor a single component, our algorithm samples from a low-energy transition path (mode-seeking behavior). When we add more components into the mixture, it starts including less likely, higher-energy paths, covering other modes and increasing the variance and mean. Note that min-max energy does not change with the introduction of several components. \n\n> Q5. Finally, I would have liked to see a discussion of the limitations of the approach. Section 6 has Limitations in the title, but does not actually discuss them in any way beyond extensions of the proposed method.\n\nThank you for the suggestion. Indeed, currently, the discussion is focused mostly on the future work rather than limitations. We will address this issue in the revised manuscript: (1) computational inefficiency in learning a mixture of Gaussian paths, (2) as already pointed out in part of our future work, the rigidity in defining states A and B to be a point mass with Gaussian instead of any arbitrary set, and (3) as also mentioned in the future work section, our method is limited to a fixed length of transition path instead of varied-length.\n\n> Q6. How do you choose the number of mixture components in practice?\n\nThe number of mixture components is a hyperparameter that should be chosen based on the complexity of the system and the available computational budget. Increasing the number of Gaussian mixtures enhances expressivity and improves the model's ability to capture diverse transition paths.\n\nWhen using learnable weights $w$, less-dominant reaction channels receive smaller weights during training. In our toy experiment in Figure 3, we observed that increasing the number of mixtures beyond the number of reactive channels resulted in only the first two mixtures having significant weights (around 0.5), while higher mixtures had weights close to zero. This suggests that the weights can be used as a proxy to determine the optimal number of components, similar to how principal component analysis identifies significant components and discards less-likely channels.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YcYLpqDWEj", "reviewer": "Reviewer_cUXK", "summary": "The paper is concerned with sampling trajectories with a terminal condition. For stochastic processes governed by a Brownian Motion, Doob's h-transform gives an posterior SDE that leads to samples with the final condition. However, estimating the h-function that is needed for the posterior SDE usually involves simulating trajectories, which is inefficient if the terminal condition is rarely reached.\nThe authors propose a simulation-free variational optimization method to estimate the h-function based on a least action principle and Gaussian approximations to the marginal densities.", "questions": "I could not totally follow the derivation in the appendix. Can you explain how to get to Equation 25 from the previous equation in line 548?\nI do not see why second term in equation 25 gets subtracted, while in line 548 only the last term has a minus.", "limitations": "The authors adequately adressed the limitations of their work.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The authors provide a solid and clear background on Doob's h transform that gets supported by the provided proofs in the appendix.\n\n2. The paper clearly highlights the challenges of the optimization problem and proposes an efficient solution adressing these challenges.\n\n3. The related work section gives a good overview and nicely connects to related topics.", "weaknesses": "1. I found the path histograms in Figure 2 to be too cluttered. I would propose adding less samples.\n\n2. The paper misses a learning curve. It would in general be interesting to have more training details.\n\nMinor Weaknesses:\n\nIn Chapter 3 and in the appendix, the authors change from trajectory length T to the unit interval. There should be a sentence that explains this change.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. I found the path histograms in Figure 2 to be too cluttered. I would propose adding less samples.\n\nThank you for your feedback and the concrete suggestion. Our goal was to illustrate the diversity of the ensemble of transition paths. We will revise Figure 2 by reducing the number of paths and will highlight example trajectories in different colors. This will help visualize the converged behavior while allowing for the investigation of individual transitions.\n\n> Q2. The paper misses a learning curve. It would in general be interesting to have more training details.\n\nWe agree with you and have uploaded a PDF containing training curves to provide more insight. While the loss itself may not be very revealing, we showcase the quality of paths (i.e., max energy) at a certain compute budget (i.e., number of potential evaluations). \n\nAdditionally, we will include two algorithms in the revised manuscript to clarify details on the training and inference. These algorithms are also included in the rebuttal PDF.\n\n\n> Q3. In Chapter 3 and in the appendix, the authors change from trajectory length T to the unit interval. There should be a sentence that explains this change.\n\nThank you for pointing out this inconsistency. We have corrected this error by unifying the notation throughout the paper to consistently consider trajectories of length $T$.\n\n> Q4. I could not totally follow the derivation in the appendix. Can you explain how to get to Equation 25 from the previous equation in line 548? I do not see why second term in equation 25 gets subtracted, while in line 548 only the last term has a minus.\n\nAfter the substitution of (24) into the equation in line 548, the third term becomes\n$$\\int dtdx\\ s_t \\langle\\nabla,q_t(b_t + 2G_t\\nabla s_t)\\rangle = \\int dtdx\\ s_t \\langle\\nabla,q_t b_t\\rangle - 2\\int dtdx\\ q_t\\langle\\nabla s_t,G_t\\nabla s_t\\rangle,$$\nwhere the last equality is due to the integration by parts. The last term in this equation, together with the first term in line 548, yields the second term in equation (25).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We would like to thank all the reviewers for their diligent review and valuable comments, which helped us to improve the manuscript. In this general response, we would like to address shared concerns raised by more than one reviewer.

To make the implementation and training procedure for our method clearer, we have included pseudocode for the training and inference processes in the revised manuscript. We hope that this addition makes the approach more accessible and easier to follow. We have also uploaded this information as a PDF in the rebuttal.

Further, some reviewers expressed interest in the training behavior (e.g., loss, training curves) of our model. To address this, we have included additional plots showcasing the model's performance in the manuscript and have uploaded these as a PDF in the rebuttal. The figures provided are:
* Loss vs training iterations.
* Transition states (i.e., maximum energy) of newly sampled paths vs number of energy evaluations.

We are providing individual responses to the questions raised by each reviewer.

Meta-review: This is a strong manuscript that proposes a novel variational technique for Monte Carlo sampling of random dynamical systems which is justified theoretically as well as experimentally. The reviewers are unanimous in its technical merit both conceptually and in practice. After the authors make a few improvements to the manuscript in terms of presentation and references, it will be an excellent addition to the NeurIPS conference program.
Decision: Accept (spotlight)