FEAT: Free energy Estimators with Adaptive Transport
Authors: Yuanqi Du
Jiajun He
Francisco Vargas
Yuanqing Wang
Carla P Gomes
José Miguel Hernández-Lobato
Eric Vanden-Eijnden
Venue: NeurIPS 2025 poster ()
Keywords: escorted Jarzynski equality
free energy calculation
stochastic differential equation
Monte Carlo method
Primary area: probabilistic_methods

Abstract: We present Free energy Estimators with Adaptive Transport (FEAT), a novel framework for free energy estimation---a critical challenge across scientific domains.
FEAT leverages learned transports implemented via stochastic interpolants and provides consistent, minimum-variance estimators based on escorted Jarzynski equality and controlled Crooks theorem, alongside variational upper and lower bounds on free energy differences.
Unifying equilibrium and non-equilibrium methods under a single theoretical framework, FEAT establishes a principled foundation for neural free energy calculations. 
Experimental validation on toy examples, molecular simulations, and quantum field theory demonstrates promising improvements over existing learning-based methods.
Our PyTorch implementation is available at https://github.com/jiajunhe98/FEAT.

FEAT: Free energy Estimators with Adaptive Transport
Yuanqi Du & Jiajun Hel Francisco Vargas Yuanqing Wang", Carla P. Gomes" José Miguel Hernández-L.obtto Eric "anden-Fijnden" 4.5
Universitty of Cambridge, *Coreell University, "aaira Therapeutics, MML Lab, Capital Fund Management, CCourant Institute of Mathematical Sciences, NYU
Abstract
We present Free energy Estimators with Adaptive Transport (FEAT), novel frame- work for free energy estimation-a critical challenge across scientific domains. FEAT leverages learned transports implemented via stochastic interpolants and provides consistent, minimum-variane estimators based on escorted Jarzynski equality and controlled Crooks theorem, alongside variational upper and lower bounds on free energy differences. Unifying equilibrium and non-equilibrium methods under a single theoretical framework, FEAT establishes principled foundation for neural free energy calculations. Experimental validation on toy examples, molecular simulations, and quantum field theory demonstrates promising improvements over existing dearning-based methods. Our Py PyTorch implementation is available at https:://// //ti://iiiii thub. com/ commjjajunheeeeeeeeeaeaaeaeeeeee jajunhe98/FEAT.
Introduction
Estimating free energy is fundamental across machine learning (appearing as normalization factors and the model evidence), statistical mechanics (partition functions), chemistry, and biology [Chipot and Pohorille, 2007, Lelièvre et al., 2010, Tuckerman, 2023]. The free energy expressed as:
F log Z, Z = (xp((33((x)))
(1)
J2
where Ω R U combines the Boltzmann constant kn and temperature T.
the energy function, assumed to such that Z < cc, and 8 = 1/knT
Rather than calculating F directly, one typically estimates the free energy difference between systems (or states) S, and Ss with energies U, and Us. which is essential for biological conformational changes, grgand-macromoleenn binding, and chemical reaction mechanisms [Wang et al., 2015]:
AF F,
Fa k log( 7 7a)
This computational challenge has driven numerous approaches. Zwanzig [1954] reformulated the problem as importance sampling, where one system serves as the proposal, enabling free energy difference estimation via Monte Carlo sampling. This free energy perturbation (FEP) method, however, suffers from high variance when the energies U, and Ub of systems Sa and S, differ significantly, particularly in high-dimensional spaces.
To mitigate this issue, targeted FEP [Jarzynski, 2002] learns an invertible mapping between two distributions to increase their overlap. From complementary angle, Bennett [1976] generalized
"The authors contributed equally this work. The order randomly assigned randomly reshufiled in different version of the paper to reflect this equal contribution. Corresponding to <jb23836cam. ac. uk? <yuanqiduech du@cs. cornell. edu>, and <eve20cims. nyu. edu>.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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FEP to create a minimum--varianee free energy estimator- the Bennett acceptance ratio (BAR) method. However, BAR still requires sufficient distribution overlap. Shirts and Chodera [2008] addressed this limitation with the multistate Bennett acceptance ratio (MBAR), introducing multiple intermediate system to improve overlap. Thermodynamic integration [TI, Kirkwood, 1935] takes a different approach by constructing a continuous path of systems S, with energies U,, with t E [0, 1], connecting S₂ and Sg. The free energy difference is defined by integrating instantaneous energy differences along this path using samples from each intermediate systems S, S:ffffectively performing infinitely many consecutive FEPs between infinitesimally close distributions.
The methods above rely on equilibrium averages, requiring exact samples drawn from the distributions of each considered state. Jarzynski equality [Jarzynski, 1997] offered breakthrough alternative based on non- eon-eqqillbrium trajectories. Similar to TI, methods based on this equality utilize a path of intermediate systems St, but only require exact samples from one endpoint, S. or Sb, allowing the law of the transported samples to deviate from the law of the intermediate systems. The escorted Jarzynski equality [Vaikuntanathan and Jarzynski, 2008] further refined this approach by introducing additional control that reduce estimator variance by constraining these deviations.
Recent advances have leveraged the capacity of neural networks approximate high-dimensional distributions for improved free energy estimation. In Targeted FEP approaches, researchers have developed invertible maps using normalizing flows [Wirnsberger et al., 2020] and flow matching [Zhao and Wang, 2023, Erdogan et al., 2025]. In the context of TI, Máté et al. [2024a,b] introduced porgyyparrmeetrrieed diffusion [Song et al., 2021] and stochastic interpolant models [Albergo et al., 2023] to capture energy interpolants between endpoint distributions.
Despite recent advances, non-equilibrium approaches and their connections other methods remain under-explored in the deep learning landscape for free energy estimation. Our work addresses this gap by introducing Free energy Estimators with Adaptive Transport (FEAT). Using samples from both endpoints, FEAT constructs non-equilibrium transport via stochastic interpolants. This learned transport then leveraged through the escorted Jarzynski equality and Crooks theorem to obtain both variational bounds and a consistent, minimum-variance variance estimator for free energy differences.
FEAT capitalizes on the key advantage of non-equliibrimm transport: eliminating the need for exact samples at intermediate distributions, thereby enabling more efficient computation. It facilitates faster numerical simulations without costly divergence evaluations while demonstrating enhanced robustness to discretization and network learning errors. Notably, our framework subsumes existing equilibrium-basee based methods as special cases, revealing larger design space with greater flexibility and performance potential. Experimental results confirm that FEAT significantly outperforms leading baselines, including targeted FEP and neural thermodynamic integration.
2 Background and Motivation
Here summarize key free energy estimators most relevant to approach. While this material is well-established in the computational physics literature, it may be less familiar to machine learning audiences. For the reader's convenience, we provide more comprehensive discussion and derivations of these results in Appendix B. For simplicity, hereafter, we set the combination of Boltzmann constant and temperature ka T 3-1 1 to absorb it into the definitions of the potential and free energy.
2.1 Free energy perturbation and Bennett acceptance ratio
Free energy perturbation (FEP, Zwanzig, 1954] estimates free energy differences between systems Sa and S, through importance sampling, using samples from one system as proposals for the other. This gives the following expression for the free energy difference AF:
AF = -log 7b/ Za) Zw) = - log E., exp(Ua - Ub)],
(3)
where we use E, to denote the expectation with respect to the equilibrium distribution (a(rr) Ze -r of system Sg.
Bennett acceptance ratio [BAR, Bennett, 1976] extends FEP with a minimal variance estimator (specifically, minimal relative mean squared error):
E. [DU - U. c
AF log(Zo/Za) - -log
+
(4)
Eg [ol - Ub + CC]

---

where Eg and E, denote expectations with respect to the equilibrium distributions of systems Su and St. and 6(x) is any function satisfying ((z)/0(-x)
e=: the usual choice is the Fermi function ((x) = 1/(1++ e"). The constant C' can be optimized to minimizes the variance of the estimator: this gives C
AF. Since AF is unknown initially, in practice it is determined iteratively by updating C to AF, where AF is computed from Equation (4) in the previous iteration.
Both FEP and BAR rely on importance sampling and become ineffective when the distributions of systems S. and S₁, have minimal overlap. Targeted FEP [Jazzynski, 2002] addresses this limitation by designing an invertible transformation T that maps samples between states. The free energy difference is then calculated through importance sampling with change of variable:
AF log E. (exp(-+)),
((x) = Un(T(x)) Ua( log VT(2))
(5)
where (z) denotes the Jacobian matrix of the map I" and [V(()) its determinant. This approach naturally integrates with neural density estimators. Recent works have implemented this transfor- mation using normalizing flows [Wirnsherger et al., 2020], computing the Jacobian via an invertible network, and flow matching [Lipman al., 2023, Zhao and Wang, 2023, Erdogan et al., 2025], computing the Jacobian via the instantaneous change of variable formula.
2.2 Jarzynski equality, Crooks fluctuation theorem, and their escorted variations
Let U, be a smooth energy interpolating between U Ua and Ut=1 Us. and consider the stochastic process governed by the Langevin equation over this evolving potential:
dX, (X,) v20, dB,,, Xo Pa:
(6)
where o, 0 is the volatility Xo Ma indicates that Xo is sampled from the distribution dz, and the arrow over the Brownian motion B, indicates that this equation must be solved forward in time. Because Ue is dime-dependent, the law of X, is not the Gibbs distribution associated with U,. Yet, Jarzynski equality [Jarzynski, 1997] shows that for these non-equilibrium effects and relate the (equilibrium) free energy difference to the work W:
Mo (dr) Z d-,
correct
AF log (ep/ex((- W)], W(X)= - a.U,(X,)dl
where Ex denotes expectation over the path measure F' of the solutions to Equation (6). Note that, we set ag = 0 in Equation (6), the samples do not move so that we have X, = Xo ~ Pa and WX++(XX- [ (X,)dt = Ux() (Xo)), so that Equation (7) reduces to Equation (3). The interest of Equation (7) is that also works when o; > O.
We can also express and interpret Jarzynski equality through Crooks fluctuation theorem [CFT, Crooks, 1999]. Specifically, consider the following backward SDEs with path measure F.
dX, = of (X,)))d + V20, aB, X1 ~ pbb,
(8)
where X1 ~ H4 indicates that X2 is sampled from the distribution (o = Zí e-t())) dr and the arrow over the Brownian motion B, indicates that this equation must be solved backward in time. Assuming that o; > O, the Radon-Nikodym derivative between the path measure of forward and the backward processes solutions to Equation (6) and Equation (8), respectively, can be expressed in terms of the free energy difference and the work as
dF
(X) exp(-W((X))))))ee AF)
dP
Jarzynski equality can be recovered from this expression by noting that its expectation over F is 1. Vaikuntanathan and Jarzynski [2008] add a control term e, the drift in Equation (6), whose aim is to better align the law of X, with the Gibbs distribution associated with U::
d = V + (x(X,)tt + V2a, dB, Xo ~ pla
(10)
Let Toe be the path measure of the solution to this SDE and define the generalized work W as:
w*(X)= = / (-V vx(x.) + VU.(X.) vx(X,) (xU((X.)) dt
(11)
3

---

where V. represents the divergence operator, i.e., trace of Jacobian.
Escorted Jarzynski equality expresses the free energy difference in terms this generalized work as:
AF = log p. (exp(exx")
(12)
This expression remains valid if we use d; = 0 in Equation (10) and it reduces the ODE
dX, = (X,)dt, Xo ~ Ha
(or = 0)
(13)
By chain rule, we have (X,) u, (X,, + (X,, (d)))) and hence the generalized work in Equation (12) becomes ( - - / V this expression becomes an implementation of Equation (5) in which we construct the map T via solution of the ODE (13) by setting T(X))) X and hence log [ So V uy (X;) This ODE-based mapping is also known as the continuous normalizing flow [CNF, Chen et al., 2018J. We will come back to this connection in Section 3.5.
Ux(Xi) Un (Xo). In this case,
can also be shown [Lelièvre et al., 2010, Heng et al., 2021, Arbel et al., 2021, Vargas et al., 2024, Albergo and Vanden-Fijnded 2024] that the law of the solution to Equation (6) is precisely (xzx) Z Ux((1) if and only if (x) satisfies
V vx(x) - VU+(x) (x) + 0,U1(z)= = F,FF
F₁ = = log Ze.
(14)
In this case, the generalized work defined in Equation (11) is determinitic and given by W*XX ln o,F,dt AF, and hence the escorted Jarzynski equality becomes a practical way to implement thermodynanic integration (TI). We elaborate on this connection in Section 3.5.
When o, > O, we can also establish the controlled Crooks fluctuation theorem [Vargas et al., 2024] by considering the backward SDE:
dX2- = of + (Xt + V2o, dB, X1 ~ Mb
(15)
Denoting the path measure of the solution this SDE as P", we have:
di.
(X) = exp(-W" AF)
(16)
The expectation of this expression over To 1, which recovers Equation (12).
3 Methods
To leverage the escorted Jarzynski equality effectively, we need a control term ii that enables Equation (10) to transport samples from S. to S, accurately. Recent neural samplers [Vargas et al., 2024, Albergo and Vanden- Eijjnenn, 2024] approach this by first defining an energy interpolant (e.g.. U, = (1- t)U,, tU,), then optimizing neural network to learn o, in Equation (10). These methods address the challenging scenario where only samples from one endpoint are available, making their performance sensitive to the choice of interpolating energy U2 [Syed et al., 2022, Máté and Fleuret, 2023, Phillips et al., 2024] and requiring Langevin dynamics trajectory simulation during training.
Our setting is different: like BAR and other established methods, we assume that we have access to samples from both systems S, and St, and our goal solely to compute the free energy difference between these endpoints. This simplifies matters and renders the specific choice of energy interpolant U2 less critical. In particular, it allows us to leverage stochastic interpolants framework [Albergo and Vanden-Eijnden. Eijnden, 2023, Albergo et al., 2023] to develop efffccint and scalable method for simultancously learning the transport between the two marginal distributions and the associated energy function U, This learning- based strategy is explained next and summarized in Algorithm 1.
3.1 Learning a Transport with Stochastic Interpolants
Given samples from systems S, and S., stochastic interpolants [Albergo and Vanden-Fijnden, 2023, Albergo et al., 2023] provide simple and efficient way to effectively learn a transport between these states in the form of Equation (10). We first define an interpolant between endpoint samples:
I = agTa + B, N65,
Da ~ Par
Ib ~ Pb: ~ (o Id)
(17)

---

where ag = 11,00 01 = 0; Bu = 0, = 1; and To ?1 0 ensure proper boundary conditions:
1==0 za and --11 It. From the results of Albergo et al. [2023], we know that the law of 4 is, at any time / E [0. 1). the same as the law of the solution to Equation (10) if we use
(18) where the dot denotes the time derivative and B1-1 _r] denotes expectation over the law of I conditional on 1, = 2. Using the L formulation of the conditional expectation, we can write objective functions for the function 2n and VU, defined in Equation (18); if we parametrize these functions neural networks of" (xx) and U/ (z)). depending on both and x, this leads to the losses:
vx(x) =[ s|, VU,(2) " F x]
(v) = 1), Ri [A luy (It) - i.]
(19)
CDSM(0) (0) = (0001 Br, Fu. [Te VU (Tt) -% -
(20)
where DSM stands for denoising score matching [Vincent, 2011], and X and are weighting functions balance optimization across different times. In practice, X and 1/t = 7 work well.
Once we have learned e," and VU we can these functions in any of the estimators presented in Section 2 via computation of the generalized work in Equation (11) or the forward backwdd Radon-Nikodym derivative (FB RND) in Equation (16). Note that, Equation (11) requires Ü,U, necessitating an energy parrgyg-aammetriieed network U². which is known to be difficult to learn via score matching [Zhang al., 2022]. In contrast, the FB RND form only requires the score function, allowing direct parameterization of VU? as a score network. In fact, using FB RND formulation also offers additional benefits, which we will discuss in Section 3.4.
We stress that the estimators presented in Section 2 are asymptotically unbiased even if we use them with functions uy" and U that have been learned imperfectly, as long as we satisfy the boundary conditions U = Ug and _ = U. In practice, however, imposing these boundary conditions puts constraints on the neural network used to parameterize U or VUţ, which may limit its expressivity and impede training convergence. For these reasons, in FEAT we choose to not impose the boundary conditions by the parameterization design, but rather learn U.² the endpoints as well. To this end, we enhance the denoising score matching loss in Equation (20) with target score matching [TSM, De Bortoli et al., 2024]:
fV TSM. (0) = (0.000 E. Fu.E [VU (T+) - of VV. (2a))
(21)
CTSM. 1 (0) (00.),) mbre (VU (4.) - vUs(xs)
(22)
This objective formulation was introduced by Máté et al. [2024b] to improve energy estimation accuracy neural thermodynamic integration (TI). Importantly, TSM does not increase target energy evaluation costs, as the gradients (Ua and (x)) can be precomputed and stored during molecular dynamics (MD) simulations alongside collected samples.
Unfortunately, even though TSM largely increases the accuracy of the boundary conditions, error still exists due to imperfect learning. Next, we discuss how to generalize the escorted Jarzynski estimators in Section 2 to account for the mismatch between U. and U, and U, at the endpoints.
3.2 Estimating the Free Energy Difference with Escorted Jarzynski Equality
Suppose that we have learned transport with stochastic interpolants for which the boundary conditions are not necessarily satisfied, meaning Ug # Ua and U o Ug This boundary mismatch requires correction term, specified by the following result:
Corollary 3.1 (Escorted Jarzynski Equality with imperfect boundary conditions). Given of and Uh. consider the forward and backward SDEs:
dX,- = VU? + of (X.)dl + 220, aB Xo Pas
(23)
dx,- of (XVU)(X,)d uf (X.)dtt + V2a, AB, X1 pb,
(24)
where at 2 O and Pha and μs denote the distributions associated with the energies Ua and Un. respectively. Define also the "corrected generalized work":
S(¹ (0x0)=/"" (-v. ( + (x.)(())) dt log exp(-Ux(X1)) (xp(-Ug((x)) (25)
exp(-U.(Xo)) expl p -U(())))
Wc(X)
correction term
5

---

Using the generalized work with correction, we have the same escorted Jarzynski equality as before:
AF - -lg (exppe/e((p_">""e log [e.(xx((W""))
(26)
where To and Te are the path measures over the solutions to Equations (23) and (24), respectively.
The proof of this proposition is given in Appendix C.3. Note that the corrected generalized work in Equation (25) coincides with the generalized work in Equation (11) if U = U, and U = Un. but
stress again the proposition remains valid even if these equalities do not hold.
The escorted Jarzynski equality in Equation (26) can be used to estimate the free energy difference:
Denoting by (1:N) ~ F" and (1:N) I" N independent realizations of the solutions to the forward Equation (23) and the backward Equation (24), respectively, we have
AF ~ - log 1/N Σx= exp(-W" (" (m)) Re log 1/ exp(W" (1))
(27)
and these expressions become unbiased in the limit as N + oc. These finite sample size estimators coincide with the apportance-weighter autoencoder [IWAE, Burda et al., 2015], and they give us bounds on the free energy difference in expectation:
E log 1/N E exp(WW (X)m)) s AF < E log 1/v En exp( exp(-W"(XXr))
(28)
These bounds are much tighter in general than the usual variational bounds:
B5. ["] S AF S E W
(29)
which are also known as the evidence lower and upper bounds (ELBO and EUBO) in variational inference [Blei al., 2017, Ji and Shen, 2019, Blessing al., 2024] and were applied to free energy estimation by Wirnsberger et al. [2020], Zhao and Wang [2023] in their targeted FEP. We prove the IWAE bounds and the variational bounds in Appendix C.I.
3.3 Minimizing Variance with Non-equilibrium Bennett Acceptance Ratio
Equation (27) provides two estimators for the free energy difference. While simply averaging them can reduce variance, we can achieve optimal variance reduction using an approach similar Bennett's Acceptance Ratio [BAR, Bennett, 1976].
Proposition 3.2 (Minimum variance non-equilibrium free energy estimator). Let W be the work terms defined in Corollary 3.1. The minimum-variance estimator is given by:
N AF = log L- ol- + + C, xa:N o (("(( -
{(EN) ~ NV
(30)
Σn'
where & is the Fermi function (0z) = 1/(1 + exx(2))). In addition, the minimum variance estimator obtained with C AF.
In practice, initialize C' as the mean of Equation (27), then iteratively update 4F using C set to the current estimate until convergence. This estimator was originally introduced by Bennett [1976] for equilibrium averages, with con-equilibrium variants later developed by Shirts al. [2003], Hahn and Then [2009], Minh and Chodera [2009], Vaikuntanathan and Jarzynski [201 1] using work likelihood or "density of trajectory" concepts. In Appendix C.2, we provide alternative derivation based on the Radon-Nikodym derivative between path measures.
3.4 Improving Accuracy and Efficiency of Free Energy Estimation with FB RND
We now turn our attention to estimators of the free energy difference using forward-backward backward Radon-Nikodym derivative (FB RND), which is enabled by the (controlled) Crooks fluctuation theorem. This approach allows us to address an issue we have left open: practice, numerical integration of Equations (23) and (24) requires time discretization, introducing additional error. We demonstrate below that FB RND-based estimators yield asymptotically unbiased estimates of free energy differences, even in the presence of this discretization error.

---

We first describe the calculation of the "corrected generalized work" with discretized FB RND: let us discretize the time interval [0, 1] into M steps to < ti <... -M- < tM 1 with step size
At, and denote by N and N the transition kernels under Fuler-Maruyama discretization for the forward and backward SDEs Equations (23) and (24):
( teta (x,) - (Nest1 Xx, o (X,, A oi (X,,)
(31)
N (X (Xe,) -N (Xe,_IXe,, ut (X., )t न.
20 21)
(32)
The "corrected generalized work" in the FB RND form can be calculated as:
dF"" (X) ~ log exp(-U6(XI)) I= .N- (X.,)
(X (
(x(X) = AF log di
(33)
Even though the quantities at the right hand-sides are only approximate expressions for those at the left hand-side, they give consistent estimators for the free energy:
Proposition 3.3 (Discretized controlled Crooks theorem with imperfect boundary conditions). Ler
N and N be as in Equations (31) and (32), and define the forwarddbuckward discretized paths:
Ne.. = XT, - of VU (X,,)t,, + 2 (X At, + V2 Th.
Xo ~ Has (34)
Xt.1 = -X,, of VUR (X At; U/ (X, At + 10 ni:
X1 pbb, (35)
where assume that o 0, Al; 1i+1 - 4i and 1/1 N(O, independent for each i = O,..., M. Then, we have:
expl U.
N (
expl U. (Xo))
(X., X... expl (1)) I. N- (X.... [x.
AF log
log
(36)
expl (U)))
II MM (X.........
The proof is given in Appendix C.3. This proposition shows that the FB RND yields estimators that asymptotically unbiased in the limit of infinite sample size, even when the time-step size is finite. Furthermore, imperfect boundary conditions do not alter the form of the FB RND; Equation (36) hold whether U/ U.. U Us Up + Ua. U A Up. This results in more compact formulation for calculating the generalized work compared to Equation (25).
The FB RND approach also offers another advantage in terms of computational efficiency as allows direct parameterization of the score VU without divergence calculations, eliminating backpropagation needs during training and sampling, thus enhancing efficiency and scalability. In contrast, the calculation based on Equation (25) requires both time derivative U,U! and divergence V V. complicating training and sampling.
In summary, to estimate the free energy difference between two systems S. and Sp. our approach learns stochastic interpolants using their samples. This yields forward and backward SDEs, as detailed in Equations (23) and (24), with potentially imperfect boundary conditions as discussed, which approximately transport samples between the two states. We then map samples from both states using these SDEs, and compute the "generalized work" via FB RND as in Equation (33). This computation results in a consistent, minimal-varianee estimator for the free energy difference as defined in Equation (30). Since our method relies on the escorted Jarzynski equality with learned escorting term adaptive to the data, we dub it the Free energy Estimators with Adaptive Transport (FEAT). An outline of FEAT is provided in Algorithm 1.
3.5 Limiting Cases of FEAT with Connections to Other Approaches
Our algorithm generalizes several existing approaches and reduces to them under specific conditions. We illustrate these relationships in Figure and elaborate below, focusing on connections beyond the already established link to the (escorted) Jarzynski equality.
FEP and Target FEP. Our method generalizes targeted FEP with flow matching [Zhao and Wang, 2023] when setting o, = O. Specifically, escorted Jarzynski equality becomes equivalent to targeted FEP with instantaneous change of variables [Chen et al., 2018] in this deterministic limit, even with imperfect boundary conditions as Corollary 3. We show this equivalence in Appendix C.4.
Taking further reduction, our method also generalizes standard FEP when both o, 0 and v 0.
7

---

Equilibrium
Non-equilibrium
BAR
0.
Jarzynski equality
FEP
=0
0, perfect boundary
FEAT (ours)
Target FEP
Pt
# perfect boundary
perfect F/
Escorted Jarzynski
TI
Figure 1: Connection between our proposed algorithm and other free-energy estimation approaches.
Comparative free energy difference (AF, unitless) estimation using targeted FEP [Zhao and Wang, 2023] and our proposed approach. MBAR results serve reference values for LJ and ALDP systems. We report the mean and std across three runs with different seeds. (*) indicates prohibitively expensive computation due divergence operations.
Table 1:
GMM (16 modes 40 modes)
IJ (w/o w. LJ interaction)
ALDP-S
ALDP-T
Method
x
x
x
Reference
234.77 #0.09
357.43
+5.43
595.98
+0.88
29.43 =0.01
-.25 +0.05
TargetFFP (FM)
0.09
-17.96
232.06
8
29.47
-4.78
FEAT (ours)
0.04 20.04
-5.34 +1.62
232.47 #0.15
356.74
#0.79
595.04 =s
29.38 20.04
-456 #0.08
In this scenario, the dynamic transport vamishes completely, and we revert to simple importance sampling using equilibrium samples from both endpoints.
Bennett Acceptance Ratio (BAR). Similar to FEP, our method recovers BAR when setting both estimator in Equation (30).
o, and U, and applying the minimum-varian
Thermodynamic Integration (TI) When U. and of" are optimally learned, our method recovers TI. Specifically, this occurs when the distribution of X, simulated from Equation (23) exactly matches the density defined by energy U.. We derive this equivalence in Appendix C.5.
This connection reveals limitation of TI: the energy function must precisely match the sample distribution. In neural TI [Mâté et al., 2024a,b], the energy network must accurately capture the data density at every time step the estimator becomes significantly biased. Our approach, based on escorted Jarzynski equality which accommodates non-equilibrium trajectories, remains effective even with imperfect learning, similar to Vargas et al. [2024], Albergo and Vanden -ijndenn [2024].
3.6 One-sided FEAT
In the last section, we focus on estimating the free energy difference between two states by learning stochastic interpolant model. Notably, the same formulation also applies to estimating the abxolute free energy, by choosing one of the states to be reference distribution with known free energy, such as Gaussian. In this case, we do not need to learn both the vector field and the score independently, as they are related in closed form [Albergo et al., 2023]. In fact, in this case, the one-sided stochastic interpolant recovers a diffusion model [Song et al., 2021]. The diffusion model learns a score network sf, from which one easily recover the vector field. We refer this variant one-sided FEAT.
To estimate the free energy difference between two arbitrary states, one can also apply one-sided FEAT to each state and then compute the difference between their estimated absolute free energies. This formulation can viewed as an extension of DeepBAR [Ding and Zhang, 2021], which estimates absolute free energies using normalizing flow with the BAR equation. In contrast, our approach replaces the normalizing flow with FEAT, enabling more flexible and scalable modeling. Empirically. found DeepBAR with one-sided FEAT achieves better performance compared to using one FEAT directly bridging two states, especially on large systems. A potential reason is learning the transport from Gaussian distribution to complex target is simpler than learning directly between two different complex targets.

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850
600
eq.
550
30-
500
Reference
450
5000
10000
15000
Number
Ilenations
(a) Illustration of ALDP-T.
(b) LJ-128.
(c) ALDP-S.
Figure 2: (a) Two states of ALDP-T. Sg: ø E (0, 2.1)): Sp: Φ [-T, U 00. m); (b) (c) Estimators
with eqs. (27) and (30) and their dynamics along training in LJ-128 and ALDP-S.
3.5
reweighted
2.0
03
1.5
10-
0.5
0.0
average magnetization
Figure 3: Eight example lattice configurations. We can see that the lattices either mostly positive negative.
Umbrella samples and reweighted histogram. We denote the two umbrellas as ua and up.
Figure 4:
4 Experiments
We evaluate FEAT diverse range of systems, from toy examples to molecular simulations and quantum field theory. Detailed experimental setups and hyperparameters appear in Appendix F.
Comparison with Target FEP. We benchmark approach against targeted FEP with flow matching [Zhao and Wang, 2023] across four systems of increasing complexity: (1) Gaussian mixtures, (2) Lennard-Jones potentials with varying parameters, (3) alanine dipeptide in vacuum implicit solvent (ALDP-S), and (4) alanine dipeptide in two metastable phases (ALDP-T, illustrated in Figure 2a). For fair comparison, all methods use identical model architectures, and apply the estimator to both targeted FEP and our approach. Reference values for the last three systems are obtained using MBAR [Shirts and Chodera, 2008]. Results in Table demonstrate that our approach consistently outperforms Target FEP. Our method's advantage over targeted FEP likely stems from the latter's reliance instantaneous changes of variables, making it more susceptible discretization errors. As discussed in Section 3.4, our approach offers inherent robustness to such errors while also climinating divergence calculations for improved computational efficiency.
Comparison with Neural TI [Máté et al., 2024a,b]. We report the results of Neural TI GMM-40 and LJ-79. For the accuracy of Neural TI, it is crucial to ensure the learned energy matches the sample distri- bution along the entire interpolant path. Therefore, Máté al. [2024b] proposed to parameterize the energy network using the energy of state A and preconditioning. To have fair comparison, we report Neu- ral TI both with and without preconditioning. Details of the preconditioning design are provided in Appendix F.5. Table 2 shows that Neural TI relies heavily such problem-specific preconditioning. which constrains flexibility and can be costly when the energy evaluation is expensive.
Table Neural TI with and without preconditioning
GMM-40 13:79
WI Precood
356.9 1.8
301.2
Different estimators and training dynamics. We visualize the estimates using only forward or backward simulation in Equation (27), and the minimum-varianee form in Equation (30) throughout training for LJ-128 and ALDP in Figure 2. Our method converges rapidly both systems, with the minimum- minimum--variace variance estimator clearly outperforming the forward or backward-only estimators.
Application on umbrella sampling. A valuable application of our method is umbrella sampling for free energy surface estimation (potential of mean force) in collective variable (CV) space. Tra- ditionally addressed via weighted histogram analysis method [WHAM, Kumar et al., 1992], this

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approach requires defining a sequence of "umbrellas" by adding harmonic potentials along the CV dimension to the target potential, then sampling from these umbrellas using MCMC. To correctly aggregate samples from different umbrellas projected onto CV space, we must estimate relative free energies between umbrella potentials for proper reweighting. Our approach integrates naturally into this pipeline by efficiently estimating free energy differences between umbrella pairs.
We demonstrate this with 5 quantum field theory, also studied in Albergo and Vanden-Eijndee [2024] for sampling tasks. The variables are field configurations E 2L and we estimate the average magnetization histogram (see Appendix F.1.5 for energy details). The lattice exists in an ordered phase where neighboring sites correlate with the same sign and magnitude, creating the bimodal distribution shown in Figure
We estimate the magnetization histogram by performing two umbrella sampling runs one biased toward negative magnetization and another toward positive magnetization- -aeiioonn-neennneee combine them by reweighting according to the free energy difference estimated by our method (calculation details in Appendix F.1.5). As illustrated in Figure 4, the reweighted average magnetization distribution successfully recovers the symmetrical nature of the 4 energy.
One-sided FEAT and large-scale experiments. We evaluate FEAT on two larger jystems-alanine tetrapeptide (ALA-4) and Chignolin-to estimate the solvation free energy. The standard stochastic interpolant struggles to fit larger molecular systems, so we adopt one-sided FEAT to learn the absolute free energies of each system and take their difference. as described in Section 3.6. Leveraging the diffusion- modell design of Karras et al. [2022], our model fits both systems well. As shown in Table 3, the proposed FEAT delivers accurate predictions, demonstrating its scalability and strong potential.
Table 3: One-sided FEAT
on ALA-4 and CHIG.
ALA-4
CHIG
Reformee
107.
190.19
REAT
109.0 2.55
320,02 + 0.70
Runtime discussion. We report FEAT's inference time in Table 4. Relative to Target FEP (FM), FEAT avoids costly divergence evaluations, significantly improving efficiency.
5
Conclusions and Limitations
Free energy difference estimation between states remains fundamental challenge with wide-ranging applications, yet research in the modern machine learning context has predominantly focused on equilibrium approaches, leaving non-equilibrium methods largely unexplored. Our Free Energy Estimators with Adaptive Transport (FEAT) address this gap by leveraging the stochastic interpolant framework to learn transports that permit both equilibrium and non-equilibrium estimation of the free energy difference through the escorted Jarzynski equality and the Crooks theorem.
One caveat of FEAT is the variance of estimator can be large even with the minimum variance estimator, especially for larger-scale systems. This is because FEAT is based on importance sampling over the path space, while target FEP is based on importance sumpling in the state space. By the data-processing inequality, the overlap between two distributions will not decrease when lifted from state space to path space. Therefore, FEAT and Target FEP with flow matching may be understood as different points on a bias-variance spectrum: FEAT is asymptotically unbiased but tends to exhibit higher variance. Recent work by Schebek al. [2025] evaluated FEAT and Target FEP for condensed -hasse systems, and highlights this point with a detailed discussion.
Our current approach requires access to samples from both states of interest. Future work could explore approaches that relax this requirement, such as those proposed by Vargas et al. [2024], Albergo and Vanden-Eijnded [2024], potentially enabling free energy estimation in settings with limited sampling access. However, these sampling techniques still face notable challenges in scalability, stability, and mode collapsing. How to resolve these issues efficiently remains an open problem. Investigating the generalizability of our approach is another promising direction, potentially with transferable networks as demonstrated by Klein and Noé [2024]. We present primary demonstration on a toy example in Appendix E.3, and leave molecular systems exploration to future works.
Finally, recent work has investigated the computational complexity of the Jarzynski equality [Guo al., 2025]. Extending their analysis to our framework would provide additional insight into requirements for reliable estimation.
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Broader Impact
This work aims to estimate the free energy difference, a core quantity in studying chemical reactions, phase transitions, and biomolecular conformational changes. We expect to have positive impacts in accelerating drug and materials discovery. However, more efficient free energy estimation also carries the risk of enabling the development of harmful chemicals toxins. We therefore advocate for their responsible use.
Acknowledgments
We thank Yanze Wang for the discussions on MBAR setup and John D. Chodera for pointing us to relevant history of MBAR. JH acknowledges support from the University of Cambridge Harding Distinguished Postgraduate Scholars Programme. CPG and YD acknowledge the support of Schmidt Sciences programs, AI2050 Senior Fellowship and Eric and Wendy Schmidt AI in Science Postdoctoral Fellowships; the National Science Foundation (NSF); the National Institute Food and Agriculture (USDA/NIFA); the Air Force Office of Scientific Research (AFOSR). YW acknowledges support from the Schmidt Science Fellowship, in partnership with the Rhodes Trust, as well as the Simons Center for Computational Physical Chemistry New York University. YW thanks the high- performance computing resources at Memorial Sloan Kettering Cancer Center and the Washington Square and Abu Dhabi campuses of New York University-we supporting teams. YW has limited financial interests in Flagship Pioneering, Inc. and its subsidiaries. JMHL acknowledges support from Turing Al Fellowship under grant EP/V023756/1.
especially grateful the technical
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FEAT: Free energy Estimators with Adaptive Transport Appendix
Algorithm
23
Extended review of free energy estimation methods
23
Proofs
24
C.I
Variational Bounds (Equation (29)) and IWAE Bounds (Equation (28))
24
C.2 Minimum Variance Non-equilibrium Free Energy Estimator (Proposition 3.2)
26
C,3 Escorted Jarzynski and Controlled Crooks with Imperfect Boundary Conditions (Corollary 3.1 and Proposition 3.3)
29
C.4 Escorted Jarzynski Equality with ODE Transport
C.5 Equivalence between TI and Our Approach with Perfect Transport
32
D
SE(3)--quivariant and -invariant Graph Neural Networks
32
Additional Experimental Results
33
E.1 Runtime Analysis
33
E,2 Robustness FEAT
33
E,3
Demonstration of Transferable FEAT
34
F
Additional Experimental Details
34
F11
Systems
34
F22
Hyperparameter
36
F3
Hyperparameters and Settings for One-sided FEAT
36
F.4
Computing Resources...
37
E55
Baseline and Reference Settings
37
22

---

A
Algorithm
Algorithm 1 Free energy estimation with FEAT
Energy function of both states Ua and Ub and their samples (I:N) Max 26 learning rate The SDE solver step size At.
(1:)
Input:
ubi
Output: Free energy difference estimator AF = - log Zo/Za.
initialization:
Randomly initialize vector field uy and score network of (or energy network Uf);; training:
repeat 0 0- The Ve (Cy. [CDSM + LU
TTSM
TSM. :) 10 + me V. (Cn): until convergence;
LU
simulation:
Simulate the forward eq. (34) from X
Di (1:N), calculate W (X(()) by eq. (33); (1:N), calculate W (X(1:))) by eq. (33);
Simulate the backward eq. (35) from X=I:N) b
estimation:
initialize C' to be the average value of the estimators in eq. (27);
repeat calculate A/ with eq. (30); and set C' +- Af; until convergence.
B Extended review of free energy estimation methods
Here, we provide an extended review free energy estimation methods. This section complements the background in Section 2 by offering more details on FEP, target FEP, and BAR, along with review of MBAR and TI.
Free energy perturbation [FEP, Zwanzig, 1954] leverages importance sampling to estimate free energy difference:
4 -- log Zo
1
log
exp(-Ub(z))dr
(37)
Za
Zu
1
log
exp(-(Ub(x) Ua(x) (x))))
(38)
Zu
= 3-log logg
exp(((Us(x)) - U. (=))))
(39)
= log E.g [xx((U> Ub)
(40)
where E,, denotes expectation w.r.t. the equilibrium distribution pa (dz) Sa. and is the density of Pa-
of state
Bennett acceptance ratio [BAR, Bennett, 1976] extends the FEP equation into general form:
Zb -lo Eg [uU U, C) Es [ō U, + C)] +
AF = log Z,
(41)
where it has been proven the minimum variance free energy estimator can be obtained by setting = 1/(1 + exp(x)) and C = AF. This estimator was first proposed by Bennett [1976] without detailed proof. Meng and Wong [1996] termed the ratio estimator "bridge sampling" and proved that the BAR form minimizes MSE using Cauchy-Schware inequality. Generalizing this estimator and the proof by Meng and Wong [1996] to non-equilibrium settings, obtain Proposition 3.2.
Multi-state Bennett acceptance ratio IMBAR, Shirts and Chodera, 2008] method > 2 states, which are typically chosen as interpolants between two ends, that each adjacent pair of distributions have enough overlap. Specifically, we are interested in estimating the free energy difference between all pairs of distributions:
2, log
AF F3 - F;
(42)
Z
23

---

To this end, MBAR bases on the detailed balance between all pairs:
Fg (oiy exp(-U,) 2,y [ass exp(-U:)
(43)
Assume for each distribution i, we use N; samples In The optimal form of day is
N₃
Gij Σ 2 exp(-Ux)
(44)
This is similar BAR, where the minimal-varianc estimator of the free energy differences contains the value to be estimated. Therefore, MBAR also takes an iterative form. Making all the necessary substitutions, we obtain:
exp(-Us(Xju)) jn)) j= 2k=1 No Nx FF Uk)
N
F = log Σ
(45)
K R
Tan [2004] first generalized the "bridge sampling" framework [Meng and Wong, 1996] multiple states and derived the optimal form of a. Shirts and Chodera [2008] then combined this formulation with the BAR estimator across multiple states, leading the well-known MBAR method.
Targeted FEP [Jarzynski, 2002] method aims to design an invertible transformation that maps the change of variable
Ta Ma to ny such that fou largely overlaps with pas- This allows formula to track the free energy difference:
Ea [ex((= [r] exp(-AF); F (T(((>)) Ua(x) log (T
(46)
where VT(x) is the determinant of the Jacobian matrix of the function T. Similarly, as the transformation is invertible, we can write the reverse direction:
[x(e(((00R) exp(- ΦR(2) (x( Ub(2) log VT
(47)
Thermodynamic Integration (TT) approach also introduces a sequence distributions that connects the marginal distributions and estimate free energy difference using the following equality:
dt OF
AF
(48)
o2. 8t di
(49)
7,
J exp( --,))
aU dr 8t
dt
(50)
J expl -U.)dr
(UU
Epp
dt
(51)
0
Øt
C Proofs
C.1 Variational Bounds (Equation (29)) and IWAE Bounds (Equation (28))
We restate the bounds for easier reference:
Let W denote the generalized work associated with samples from the forward and backward SDEs:
dx,- -U of of V2a, dB.
Xo Pa:
(52)
dx, V. (x,)dt (X,)dt + 22, TB
X1 pb:
(53)
We then have
N Σ
N exp ( < AF < -E log Σ exp(-W (P(())))
E
log
(54)
N
n=1
24

---

and
W Σ AF & Ep. [W
(55)
Proof. These bounds are corollary based on the escorted Crooks theorem, which says
d dF (X) = exx==xx(- + AF)
dF
log
dFv (X)
-
(X) + AF
(56)
We first look at the ELBO and EUBO bounds, and then we prove the IWAE bounds. Taking the expectations escorted Crooks theorem, obtain
d
E
log
= Bz. 1- AF,
F log
E
AF
(57)
u
We recognize their LHS are negative KL divergences, which are always non-positive. Therefore,
E AF O, E [p.[W"" AF 00
(58)
Rearrange these equations, we obtain
E W S AF S E Ei [W",
(59)
which finishes the proof.
The IWAE bound can then be obtained by Jensen's inequality. The green color indicates the terms in Equations (28) and (29) for a clearer visualization.
AF
= log E
X~ pu
(60)
N
= log E
exp(WV (X((h))))))))
(61)
N
n=1
2 Exain
log N
Σ n=1
exp(W"(X((n))
(62)
log exp (xx((x(
> E Exa.M Fu
(63)
N
==1
EF. W
(64)
AF
- log E Ex-Fo
expl exp(-W"((X))
(65)
log E Exa:MiFt
exp(- exp(-W"(XX(m))
(66)
- E ExCiNN
log Σ exp(-W" (x(()))
(67)
n=1
log exeepp- (oge
- -ExaM N
(68)
Ep
[W']
(69)
We hence can see that IWAE bounds are tighter compared to the ELBO and EUBO bounds.
25

---

C.2 Minimum Variance Non-equilibrium Free Energy Estimator (Propasition 3.2)
Proposition 3.2 (Minimum variance non- non-equilibrium free energy estimator). Let W Wa-sb and Wo-te be the work terms defined in Corollary 3.1. The minimum-variance estimator is given by:
E O (x()) C, X(IN) Bo, X(IN) P"
AF log
(70)
Ex= (W - C)
where & is the Fermi function <(-) = - 1 exp(-)) and C AF in optimal.
Proof. Our proof follows Bennett [1976], Meng and Wong [1996] closely, with slight extension to path measures. To increase the readability of the proof, we first summarize the entire structure:
1. express the normalizing factor ratio;
2. express MSE of the normalizing factor ratio estimator, and approximate it with d-method;
apply Cauchy-Schwand inequality to obtain a lower bound of MSE? This gives an optimal condition minimize MSE?
plug the condition back to the normalizing factor ratio expression and finish the proof.
1. express the normalizing factor ratio:
First, we have the following equality:
Z,LE, a(X)g [ = ZoBp [a(X) (-W(x))
(71)
where is arbitrary function, and g is any function such that g(r)/g(-r) exp(-r). Our goal is to find form of a to minimize the variance (MSE) of the estimator of the ratio between normalization factors. Also, make things clearer, explicitly write the direction in the path measure P Pa-sb, Po-aa and note that we drop the superscript (") for simplicity.
The equality Equation (71) holds, because:
Z Ep
((X)g
[(X))]
(72)
-p
10 a(X)) (-W(X)) exp (-W(x))
b as g(r) = g(-r) exp(-r)
(73)
log dPo-e
(X) + log 2b
Z )
-ZaBr..o
a(())
(-W(x)) exp
(74)
dPa-sb
Za Zo Er
(a(X)1 (-i(x))
dFo
(X)
(75)
Za
dPa-sb
Zp. m Ta(X)1 (-W(x))
(76)
2. express MSE² of the normalizing factor ratio estimator, and approximate it with 8-method:
We now write down its Monte Carlo form. For simplicity, we assume we use the same number of samples from ((IN) ~ Pa-sh and (1:N) ~ Ph-aa
Σ a((((")))) ( (W(x(=)))) Zn N
1
20
N Σ
((((()) (n) (a )g (-
(77)
26

---

Therefore,
1
Σn
a(x (m)
b )
(( (1))
Zu
N
f
(78)
Zb
En
(a(x 1
W ((X(m)))
N
We consider the MSE of A:
MSE = E(f - -r|?//²
(79)
To simply the calculation of MSE, ket vi, 7/2 be respectively the numerator and denominator of A:
1
Σ
aa(x 19 ( (- -
f
(80)
N
1
( (
f22 N
(81)
and we denote 7) and 1/2 as the true value:
m Es (n(Xoma)s (-W -W(Xmmm))
(82)
722 Eu-s a((X-1b9 ((
(83)
where we write Ex shorthand of By.... Note that
Zom Za72
(84)
Following Meng and Wong [1996], we use d-method to approximate the MSE:
(m)
((re)
MSE = Elf r///1/
(85)
71
2
22
The variances are given by:
V(ñ) N Eb-ra a (x(b)))) (-W(X ) - -R2 [a(Xb>a) (-W(X
(86)
((f2)) = N Ea-+b a a(Xo-1)² (W(Xxs))) 27 EZ-so aa(Xa>b)g (W(X
(87)
and hence we have
a a(Xoru) ( (-W(Xo-)a)) 27
(W(Xors)) Z Ba-bb al a(xo-10 ( )²
Eg
+
23
2
MSE
133 aa(X ( (X
N
(88)
We now look at the second term in the numerator:
za
a(Xo-1))
(89)
Z3
22
E
+b
[a(Xa-b))
W (X g (W(Xa-oo)
(90)
23
Z2
Eo
[a(Xa=>)
W ((oa+bb) 9 (-W(Xo-o)) exp (-W(Xa-o))
(91)
Z2
722
dl dPo-aa (Xa-b) + llog
Zbb
Ea-+b
a(Xo-+)) (X
(-W(Xx-t)) oxp
log dla-bb
23
Za
(92)
Zu
(a(Xav)'' W ((Xa-b)) 9 (-W(Xx=s))
dlp dPpos (X dP.ob
Ba-
(93)
Zo
Zea
Es-ra
[a(Xoma)' (((X)aa) g (-W(X
(94)
Zb
27

---

Therefore, we can write the MSE as
MSE
Es-ra (a(Xo-a)2g (W(ooa)) (g (o(-W(Xn-oo) + 20 W (W(X--aa)
2
(95)
B B a(Xo-ra (-((X
N
apply Cauchy-Schwartz inequality obtain a lower bound of MSE?:
We denote
A(X) = a(x)²g (-W(x)) (1
2/2 (w(X))
Za
+
(96)
%
B(X) 9000gg(-1 (- (1 (-W(x)) +
(i(x)))
Zn
(97)
We write the denominator as
Ez [a(X6-a)9 (-W(Xom))) E2 EB [VAVB]
(98)
and the entire MSE
MSE Es-sa [A] /B VAVB 2//N
(99)
Following Meng and Wong [1996], we apply the Cauchy-Schward inequality:
E
VAVB B
s to [A] Ex-ra [B]
(100)
Due to the property of g, we can see B is always positive:
2a B(X) , 10 (-w(x)) (00))(((((0 (-W(x)))
(m(x))
+ %
(101)
= 1/(1+ + La 14, exp(e exp(-W(X))
(102)
We hence have
Eg-ra [A]/E]_4
VAVB 1/ [B]
(103)
Note that Eg [B] does not depend on a. Hence, found lower bound of the MSE w.r.t a. The equality holds when A x B, i.c.
1
((X)
(104)
(-W(X)) + La/ Zog ((X)
4. plug the condition back to the normalizing factor ratio expression and finish the proof:
Plugging back to Equation (78), we obtain
of ((-W(x(x)
Σn
Z,
9 (m) (n) +Za /Zg ((x)(()))
2
(105)
Zo
((X("))))
Σn
0 (x( +2a Zao W(x("))
28

---

We now use the property of 9. namely g(r)/g(-r) = exp(-r), to simplify both the numerator and the denominator:
9 (-
(106)
9 (-W(X
7a/Z/g
W (
(107)
1 Za/Za exp
-W(X
(108)
1 + exp((AF) exp
(-W(X
1
(109)
(-W(X AF)
1 1+ exp exp
and
[ (X
(110)
9 (- (-W(x (m))))
Za/ Zng W (X((m)))
(111)
exp (W(x((bb)
+ 7n/7%
exp(-AF)
(112)
((x(1) AF) +1
Taking log on both sides of Equation (105) and plugging in the simplified numerator and denominator, we obtain
Σn,
Zn
Ltexp|
-X(((m) )4F
AF = log
2 log
(113)
Zb
Σ
exp(-AP)
W(X(m) --00--4-- -4F)
Σn
14exp W
-(( ()) )4FF)
log
+ AF
(114)
Σ,
exp W(X - -
Let o(.) = 11(1 + exp(-)) as the Fermi function, we have
En 5 AF) + AF
AF log
(115)
Σn (W((X_)))) AF)
which finishes the proof.
C.3 Escorted Jarzynski and Controlled Crooks with Imperfect Boundary Conditions (Corollary 3.1 and Proposition 3.3)
Corollary 3.1 (Escorted Jarzynski with imperfect boundary conditions). Given of and Uh, consider the forward and backward SDEs:
dx, -0 (X.)d ( 220, dB, + /20, AB, X1 pb.
Xo Pas
(116)
dx;
(117)
29

---

where ot 2 0 and and μbb denote the distributions associated with the energies U and Ub.
respectively. Define also the "corrected generalized work":
(x) //¹ (-V. (X..) (X.)
-0 (X.) 0,0% oxU)(X)))
+ log exp( ( exp( (xp(-U(((1)) exp(-U exp( (Ugg))))))))
(118)
correction term
Using the generalized work with correction, we have the same escorted Jarzynski equality before:
AF - log Ee. log E Ep./ppp(W"
(119)
where To and Te are the path measures over the solutions to the forward and backward SDDE respectively.
Proposition 3.3 (Discretized Controlled Crooks theorem with imperfect boundary conditions). Let N+ and N be as in Equations (31) and (32), and define the forward and backward discretized paths via
X N -x,, of VU (xx;) A1, + o Al, + 22 Te.
X
pl: (120)
X Xx. of
of VU (X At;- V/g Xx t, At, V2At,-104, Thi
X Pb. (121)
where we assume that 0t > 0, At; = 11+1 4 and 1/3 ~ (1,, Id), independent for each i = O,..., M. Then, we have:
exp(-U6(X1i) i N (R.,, Xe,) exp(-Uu(Xx))
AF log E
(122)
N+ N+ (8
exp( Cup(-U,(A (X)() W+ (x
log E
(123)
exp( (Xp(- N (X X1,
We first prove Corollary 3.1.
Proof. First, we note that the escorted Jarzynski AF - log E [e.ppx((-WW") log Ep. (ex)() be obtained from controlled Crooks theorem. Therefore, to show Equa- tion (119), we only need to prove:
L. (-v us (x,) VUI (X.) + + a,U) dt
log exp( Ua (Xoo) exp( (U AF - log di.
(X) (124)
exp( U,, exp( (Xo)) (Xo))
dF
Consider the SDEs as Equations (23) and (24). However, instead of starting from Ua and Up we start from Ug and UP. We define their path measures as P.' and ie' and we have
d dF v (X)
d
exp( exp(=Ux(Xo))/7a
(xp(-U((1))/24 (xp(-U (xp(-Ug(Xo)) (Xo)) /uz
di (X)
(125)
According the controlled Crooks theorem (as in Equation (16) applied the transport between UP and Uo, we have
L. (-V. + VUL vf" + +.uUf) dt = - log
Zoz
log
d (X)
(126)
Zug
Take logarithm of Equation (125) and add with Equation (126), we obtain
L. (-V..u0 + -Uf of + +0,U?) dt + log exp(-Us(X)
exp(-U. exp( -Xp(U
(Xo))
Zo
Zyp
ZyZ
db
log
log
-log
log
(X)
(127)
Zrg
Zuz
d
AF
which finishes the proof of Corollary 3.1.
30

---

Next, we look at Appendix C.3.
Proof. We will only provide proof for Equation (122) in the following, and Equation (123) follows exactly the principle. As the Gaussian kernel in Equations (120) and (121) is normalized, we have
f exp(-U,(X) II
M N (X
(X,,,
Zb.
(128)
M
/ exp((UU N (X
Za
(129)
Therefore.
(xp(-U6(X11) exp( Expl-Uu(X o)) II
log
(130)
exp(- (X )) IT N+ (X,, Xx,.)) exp( ( II N' (X,,,
=- log
dx
Za
exp( -U, o)) II N
(131)
Texp(- (Ub((X))
(=1 N N
(X XxX
=-log
(132)
Za
Zo
log
AF
(133)
Zo
C.4 Escorted Jarzynski Equality with ODE Transport
We restate the equivalence of escorted Jarzynski equality with ODE transport and target FEP formula with the instantaneous change of variables in the following:
Let X, evolve according the ODE
dX. dí, Xo pa-
(134)
For smooth energy function U, with, potentially Un * Ua and U1 # Up. the escorted Jarzynski equation with imperfect boundary conditions (Corollary 3.1) holds:
AF -loog Exarpe Texp(-
W / (-V VU, + a.U,) di log exp(-U6(X1)) exp((UX(Xo))
(x(-UU((X))xpp(-Uiii1 expl (135)
This is equivalent to the target FEP formula with the instantaneous change of variables:
AF =- - log Exo-a [exp(- ((X) = Ux(X)) Ux(X)) / v uz dt.
(136)
We note that the proof for escorted Jarzynski by Vargas et al. [2024], Albergo and Vanden-Eijnden [2024] requires de o Concretely, the proof by Vargas et [2024] relies on the FB RND, which applies only to SDEs. On the other hand, while Proposition 3 by Albergo and Vanden-Eijnden [2024] states that o, 2 0, the proof essentially requires o, * order to eliminate o, from both sides in Equation (63) on page 19. One valid proof for
0 was provided by Tian et al. [2024] using the generalized Liouville equation. Here, we consider more straightforward derivation, which also directly showcases the equivalence to target FEP formula with the instantaneous change of variables.
31

---

Proof. To prove Equation (135), we directly show the equivalence between & and W To do so, we consider the total derivative of (x)):
dU,(
(U+(X+) (U)( dx, (U)(X,) -
(137)
dt
Öt
dt ôt
The second equality is due to the ODE Equation (134). We therefore have
W = ! (-V
Uz + VUz + +.UU) dt + log exp(=U,(Xo)) exx(U((XX))
(138)
exp(-U:(Xi)) exp((U6(XX))
(X,)
//¹ V dt log exp( (xp(-U
(139)
dt
exp( (xx(-UUU)))xpp(-uuu
exp( U, (x)))(-)))))))
-X)( Uo(Xo)
V ur dt + log
(140)
exp( (X,)) cxp((XX))))))
/ V dt
= Ub((1) - Ua(Xo)
(141)
= (X)
(142)
which finishes the proof.
L
C.5
Equivalence between TI and Our Approach with Perfect Transport
When our method is optimally trained-such that the density of samples simulated from the SDE matches the energy U2 at every time step-it it becomes equivalent to TI. We only prove this equivalence using forward work, while the backward work will follow the same argument. To show this, let denote the sample density at time step t. At optimality, following Albergo and Vanden-Eijnden [2024], we have:
V (DePt) = -.Pt
(143)
V upPt + 74 V P: Ô,pt
(144)
CPPt UpP V Us = -rOd log
(145)
04 2a V U. 0.logPt
(146)
Then, the generalized work
w S( (-V- u- VU.
VV or + 0,U,) dt
= =f (-0, log + 0,0%) dt /( -0,Z,dt = Zo - Z1 (147)
i.e., Wo will always be constant under perfect transport. Therefore, write the escorted Jarzynski as
AF log E
(148)
Es. [Wo]
D as W is a constant
(149)
/ Ep,[- log + J,U,Jdt
(150)
/ Epp |- θ lognd di / Ep (a,U,
(151)
=J (x)d(()
TI formula
D Squiieaqn and -invariant Graph Neural Networks
For n-particle problem and molecular systems, we adopt F(n)-equivariant graph neural networks (EGNN) proposed in [Satorras et al., 2021]. Given 3D graph G (V, E, where V is a set of
32

---

vertices, E VVV x V is set of edges, X E INN Nx3 is set of atomic coordinates and H E RNK is node features with feature dimension K. The procedure of EGNN the following:
rij z -
(152)
m E
(e My (lyy
(153)
JEN(()
ht ob (h/, m²)
(154)
1+1 at + E (1)))(m))
(155)
JEN()
where we discard the coordinate update when we only need SE(3) invariance. To break the re- flection symmetry, we introduce cross-product during the position update which is reflection- antisymmetric [Du al., 2022].
Gj = a 2 | x xلyl
(156)
x = o +
Σ ()))a(m)) + (m))(mm)
(157)
JEN()
where Φe. @h. Pr and &e are different neural networks to encode edge, node, relative direction and cross direction scalar features.
E
Additional Experimental Results
E.1
Runtime Analysis
Table 4: Inference time for FEAT and Target FEP (FM).
Name
GMM (d=100)
LJ-55
ALDP-T
FEAT (ours)
8
min
20
Target FEP (FM)
40
>10 h
40 min
Besides the results in Table 4, we also include brief discussion against neural TI and MBAR below:
Versus neural TI, FEAT performs transport from both sides, which can roughly double its runtime. However, we observed that Neural TI requires a much larger sample size, and preconditioning for the network to achieve ideal performance, which largely increase its inference time.
Versus MBAR, FEAT is neural network approach that requires training. But it does need intermediate samples. We here take an example using ALDP. This can make it more favorable when sampling from the intermediate is expensive. For ALDP, generating samples for each target requires about 1 day our machine. Collecting all the targets for MBAR can hence take between 1-10 days, depending on whether the simulations are run in parallel. In contrast, our training process takes only 1-2 hours, which is significantly faster than sampling from all intermediate densities. However, this comparison can vary depending how the sample collection pipeline are implemented.
E.2 Robustness of FEAT
In this section, we analyze robustness of FEAT and Target FEP with flow matching against different sample size and number of discretization steps on GMM with different dimensionalitiis. We can see FEAT achieves greater robustness toward less steps and less samples. This results also reflect discussion in Proposition 3.3 for discretization errors.
33

---

Table 5: FEAT
Table 6: Target FEP
Mstep
sample size
GMM-40D
GMM-100D
Rstte
sample size
GMM-40D
GMM-100D
50
5000
40.05 0.33
-4.04
1.92
50
5000
+0.98 + 0.23
-15.44
100
5000
0.12 + 0.07
-6.43 + 1.72
100
5000
+0.48 1 0.17
-12.10
4.80
500
5000
0.00 + 0.06
-3.59 = 1.06
500
5000
-0.12 + 0.08
-13.72
500
500
0.13 + 0.09
-5.56 + 1.87
500
500
-0.16 + 0.37
-22.24 + 3.43 3.43
500
1000
-0.04 0.05
-6.46 + 1.86
500
1000
0.02 =
-19.73 2.13 2.13
500
5000
0.00 + 0.06
-3.59 = 1.06
500
5000
-0.12 0.08
-13.72 + 2.46
E.3
Demonstration of Transferable FEAT
FEAT can be trained on several datasets from multiple targets, with conditions on the target parameters. Once trained, this model will allow us apply FEAT to similar systems without re-training, similar to what has been demonstrated in transferable Boltzmann generators [Klein and Noé, 2024].
We showcase this potential on GMM-40 with different scaling factor. Concretely, we scale the state B with scalar 0.5, 0.7, 0.9, 1.1, 1.3, 1.5 when training. The network also takes the scalar as an input. After training, we evaluate on unseen scalars and report the average eπrоr and standard deviation across 3 runs. As we can see, the transferable FEAT model achieves good accuracy across range of unseen targets. Also, expected, interpolation yields better performance than extrapolation.
Table 7: Transferable FEAT on GMM with different scalars unseen during training.
Scalar
0.45
0.6
0.8
1.0
1.2
1.4
1.55
Error
std -3.52 + 1.13
-0.07 + 0.05
-0.02 + 0.06
-0.01 + 0.06
0.01
0.04
0.003 + 0.04
0.03 + 0.13
Additional Experimental Details
F1
Systems
F.111
Gaussian Mixtures
We consider estimating the free energy differences between two Gaussian Mixtures in 40/100- dimensional space. Our implementation is based on the code by Midgley al. [2023]. Below parameters for S, and Sy:
Sa: 16 mixture components, components mean ~ 2 std softplus(-3) random seed 10.
Sy: 40 mixture components, components mean (((222)), std softplus (-2), random seed 0.
F.1.2 Lennard-Jones (LJ) particles
We consider alchemical free energy for N = 1255/9/128/ IJ particles. We obtain samples from states S, and S, with the addjuee Langevin algorithm (MALA) for 100,000 steps. We remove the first 20,000 samples as the burn-in period. The step size is dynamically adjusted on the fly to ensure the acceptance rate is roughly 0.6. Below are detailed settings for S, and Ss:
Sa: LJ-potential with the harmonic oscillator:
Un E -un((, -
X
X
(158)
N
141
where
Unl(r) = de
((]
0
67
(159)
In our experiments, we o =c
34

---

Sa: the harmonic oscillator:
U₂
Σ
Xn
Σ
X
(160)
n'=1
where X, E R is the coordinate for n-th particle in the system.
F.1.3 Alanine dipeptide solvation (ALDP-S)
consider the solvation free energy between ALDP the vacuum environment and with implicit solvent, defined with AMBER f196 classical force field. Specifically, the samples were gathered from 5 microsecond simulation under 300K with Generalized Born implicit solvent implemented in openmmtools Chodera al. [2025]. The Langevin middle integrator implemented in Eastman et al. [2023] with a friction of Ipicosecond and a step size 2 femtoseconds was used to harvest a total of 250,000 samples. The same sampling protocol was used in the following paragraph well. Below are settings for Sa and Ss:
Sa: ALDP in the vacuum environment;
Sy: ALDP in implicit solvent.
We also rescale each target scale by 20, i.c.. we define the energy U (.) Note that this will only change the scale of input and the score, with no influence on the free energy difference as long as we apply the same scaling to both targets.
F.1. Alanine dipeptide- transition (ALDP-T)
We consider Alanine dipeptide in the vacuum. As shown in Figure 2a, there are two metastable states in this system. We therefore consider the transition free energy between them. Below settings for S. and Sy:
Sa: ALDP in the vacuum environment, Φ E 20)).15):
Ss: ALDP in the vacuum environment, Ø [-,, 0] U [0, T)
Similar to the solvation case, rescale each target scale by 20.
F1155
4 Iattice field theory
For ep experiments, we consider reweighting the histograms obtained from two umbrella samplings with the free energy estimated by our approach. The random variables here are field configurations R 1. and the energy function is defined as
U(p)=
= Σ
>
PaPetn + (4 m²) + xxp2
(161)
Here, use fo to represent the value of at index We choose m² -1, 0.8 follow- ing Albergo and Vanden-Eijnden [2024]. The two states are defined with two different umbrella samplings:
Sa: UV(F) = U(p) + 2n 9x 21)2:
Sy: Ub(p) = U())+ 2 Σ. 9x 22
where k 12 10, and μ4 = - 0.3, p1 = 0.6. We deliberately choose asymmetric values, symmetric setup will render the analysis less complicated.
We then estimate the free energy difference AF F. F. with our proposed approach. With this estimate, construct the histogram of the average magnetization by reweighting the samples from U, and Ub. Concretely, for each bin & in the histogram, we compute its reweighted probability as (E) + mo(E)
P(E) N₂ exp( F. 4 p1)2) + No ex,e((FFF ' - 22)
(162)
where N₂ denotes the total number of samples from ua, and na is the number of those samples falling in bin E
35

---

F.2 Hyperparameter
We include hyperparameters for model training and evaluation in Table 8. We explain some hyperpa- rameters in the following:
Or Pe. Yx these parameters define the interpolant: X, a, + 8,X, + TcC, where NO I);
OT pair: To facilitate training, instead of randomly sampling pair (Xo, X1), we compute an optimal transport (OT) plan to select pairs of data points that are closer to each other. We use the implementation by Tong al. [2024] (MIT License) to find the OT pair.
Additionally, we note that standard OT chooses the closest pair in the Euclidean distance, which does not directly apply Lennard-Jones data. To solve this problem, we first canonicalize all data points. Specifically, we select one sample as the reference system and rotate all other samples from both states to align with this reference using the Kabsch algorithm [Kabsch, 1976]. For the Lennard-Jone system, we additionally canonicalize atom permutations by applying the Hungarian algorithm [Kuhn, 1955, 1956] to find the optimal assignment that minimizes the distance matrix between each sample and the reference. Similar approaches were also adopted by Klein et al. [2023]. In practice, we found this significantly accelerates the training and enhances the performance for larger systems.
rolation-invariaan alanine dipeptide and cermutaaioo/rroaaionnnnnnann
OT batch size: instead of running OT on the entire dataset, we run OT within a much smaller batch to ensure low running cost. Note that this number is different from the "batch size".
FM warm up: we found it is helpful to warm up the vector field network with flow matching, especially for GMM high-ddmensional space. If we use this warm-up, will put the iteration number in the table; otherwise, we will leave "-".
oy: recall that during simulation, we run the forward and backward SDEs as defined in Equa- tions (23) and (24). a, is the diffusion term in these SDEs. We note that o, is not the noise level for the stochastic interpolants (which is 7)).
Table 8: Hyperparameters our experiments.
GMM
W
ALDP-S/T d 22 x 3
Hlperrparameters
d 40
d
100
d
55
x
d
79
x
d d 128 x
Model and Interpolant choices
Network architecture
ML.P 5, 400
SE(3)-GNN
Network size
4, 64
1
82
7x
at
f), 0.05
Training
learning rate
0.001
batch size
1,000
1,000
100
30
20
500
iteration number
50,000
200,000
10,000
20,000
40,000
20,000
pair
No
Yes
No
Yes
Yes
Yes
OT batch size
1,000
500
500
500
FM warm
50,000
Simulation and Estimation
number discretization steps
500
evaluating sample size
1,000
5,000
1,000
1,000
1,000
1,000
0.01
F33 Hyperparameters and Settings for One-sided FEAT
Network. For one-sided FEAT, we only need one network to parameterize the score/mean/noise for the diffusion process. We adopt the parameterization (precisely, Cin, Cskip: Cout) following Karras et al. [2022], with an EGNN (fully-connected) as the network to prediction the mean E[X,] X.). We increase the EGNN size with 5 hidden layers, each with 256 hidden units.
36

---

Training details. We train the network for 200,000 iterations, with batch size of 20 for both Chignolin and Ala-4. We observe better convergence for larger batch size, while we choose 20 to fit in our GPU. We keep an EMA with rate 0.99.
Estimation details. We estimate the free energy for each states with 500 discretization steps. We 1,000 samples for ALA-4 and 3,000 samples for Chignolin. Additionally, we found it is more stable use DDPM kernel [Ho et al., 2020] as the denoising and noising kernel instead of EM discretization kernel. This will only slightly change the formulation of the Gaussian expression, with other key components of FEAT unchanged.
Precisely, the forward SDE is defined following Karras et al. [2022] as:
dX,- = V2d. Xo
(163)
where j is the target system distribution. The backward SDE is defined as:
dx; 2/V (X,)dt + VIdd XT ~ (²)
(164)
where VU, is the learned score network. The score and the mean- preddipeeionoon E [X]] are connected with Tweedie's formula:
-U,( X, E[Xo
(165)
1
The DDPM kernels for this pair of SDES are:
~ (Xe,i (X..) N (X1441
X1. - -1,)1)
(166)
& ( - 6,11
N (Xe,) NN [1,,
Xt + (1
EE
(167)
72 1+1
1++
E4
Computing Resources
All experiments are run on a single 80G NVIDIA H100.
E5
Baseline and Reference Settings
Target FEP. We use the same parameter as Table 8. Note that the interpolant becomes X, ayXo + Br_X,, and the simulation process will be ODE. We do not align the iteration number to be the same as SI. Instead, we run the training until convergence.
Neural TI. We use the same parameter as Table 8. We parameterize the energy network instead of the score network in order to use the TI formula. Specifically, we take the output of the network, and take an inner product with the input form scalar the energy.
Neural Preconditioiing Design. We include preconditioning for the energy network to ensure boundary conditions and also to increase the accuracy of the learned energy U..
For GMM, we set the energy network to be
U(a,, t) = u + (x.) + czU,((
(168)
where a, = cx) fo [o] (1-t), by cxp(go(1) go(()] -t and et (xo(ho(t) (t)) -t t- (1 t). for Io. hgg Ug are neural networks. We can exactly ensure the boundary condition by this.
For LJ, UA and Un is more sensitive to noisy dt. Therefore, inspired by Máté al. [2024b], we set the energy network to be
U( :) =; UnL re, az) + c Uo(2,,
where r; is the radius parameter in LJ, ranging from 0 to 1. exp(ax - (1 (1-th t) is smooth parameter, as used by Máté et al. [2024b]. b, = 1 cx)) are scalar to ensure boundary conditions.
and e = cxp()o(t))---11)))
Due to this specific choice of smoothing parameters, we failed to design a stable preconditioning for ALDP and hence did not compare FEAT with neural TI on ALDP.
MBAR. We use MBAR (with pymbar) to obtain the reference value for the LJ system and ALDP:
37

---

LJ: for LJ L-potentia,, we create N distributions as follows:
N
N
U. = Σ
X
1
Σ
X
(169)
N m²==1
$5
n=
where
UL)((T) 4c (
12
(170)
We set 01 =1 NN cN = and let on Cn vary linear interpolant between 1 and 0 For LJ-55/79, we use = 41 distributions, and for LJ-128, we choose = 81. We run the Metropollis-djussted Langevin algorithm (MALA) obtain 20,000 samples for each marginal, and then randomly choose 1,000 of them for each marginal to decorrelate the samples.
ALDP-S: set distributions by modulating the solvation parameter with a factor À E [0, 1] that scales the charges the "Custom(BForce' in the force field with OpenMM. Specifically, modify the force by the following code:
system getPorces
if force. -la......aa..................... _naee....._._...... == CustanGBForce':
for 1dx range (force. Carree.getuunParttiee () charge, signs, epsilon force. sesParticleParmeeteers (idx) force. aarre..otParticeePaaaacaa. (idx, (chargo+lanb, signa, opsilon))
Listing I: Python code example changing the solvation strength.
The 11 distributions are set with = 0.0, 0.1...., , 1.0. We run MD for each distributions using the setup described in Appendix F.1.3, and randomly choose 2,000 of them for each marginal to decorrelate the samples to run MBAR.
ALDP-T: we define three distinct distributions: (1) distribution containing only Sa, where the energy in regions corresponding to state Sp are set to +00, (2) a distribution containing only Sp. where the energy regions corresponding to state S. are set to +00; and (3) full distribution that includes both S, and S..
38


Reviews:
[{"review_id": "MyJ4hblSgD", "reviewer": "Reviewer_ywwp", "summary": "In this paper, the authors employed an adaptive transport method called FEAT to predict free energy profiles for chemical reactions. FEAT can handle both equilibrium and non-equilibrium samples. FEAT was also applied to umbrella sampling analysis, replacing the traditional method WHAM.", "questions": "1. Could the authors include more real-world case studies (beyond toy systems) for free energy prediction?\n2. Could the authors provide details on the training sets used for the alanine dipeptide (2-state), Lennard-Jones (LJ), and ALDP-S systems? Additionally, please clarify the differences between the \"2-state alanine dipeptide\" and \"ALDP-S\" systems\n3. What are the limitations regarding the applicability of the FEAT method to explicit solvent systems or larger (e.g., biomolecular) systems?", "limitations": "The scope of evaluation appears limited to small model systems, with no comparative benchmarks against some MD driven systems like drug-target systems.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 4, "strengths_and_weaknesses": "**Strengths**:\n\n1. This work presents an effective application of deep learning models for free energy calculation, a relatively novel area within AI4Science research.\n2. The method demonstrates application in several examples with real-world scenarios.\n\n**Weaknesses**:\n\nSee limitations", "was_revised": true, "final_justification": "Authors' rebuttal has address all my concerns.", "rebuttal": "Thank you for your valuable review. We are pleased that you found the method in our paper effective and novel. We have addressed your concerns with new experiments on larger systems and detailed clarifications about our experimental setup. We are committed to addressing any remaining concerns you may have. Should you find our reply satisfactory, we kindly invite you to reconsider your rating. \n\n## Responses to Questions\n\n> ### Q1: Could the authors include more real-world case studies (beyond toy systems) for free energy prediction?\n\nThank you for this important suggestion. We provide FEAT's results on two larger systems: **Alanine Tetrapeptide** (polypeptide with 4 amino acids) and **Chignolin** (protein with 10 amino acids).\n\n**Alanine Tetrapeptide Results:**\n- System size: 43 atoms (129 dimensions)\n- FEAT estimate: **65.51 ± 1.51 kcal/mol**\n- Reference value (MBAR with 20 intermediate distributions): **64.09 kcal/mol**\n\n**Chignolin Results:**\n- System size: 175 atoms (525 dimensions)\n- For this larger system, we found that it is more effective to decompose the estimation task: instead of directly learning a transport map between the vacuum and solvent states, we estimate the free energy using FEAT in two stages—from a Gaussian to vacuum, and from a Gaussian to solvent—and then take the difference between the two estimates. \n- FEAT estimate: **190.784  +- 0.414 kcal/mol** (due to time limit, we did not train 3 times, and this std is obtained by doing 3 inferences with one model.)\n- Reference value (MBAR with 20 intermediate distributions): **190.887 kcal/mol**\n\nThese results demonstrate FEAT's applicability to larger, more realistic systems. We will update these results in our camera-ready version.\n\n> ### Q2: Could the authors provide details on the training sets used for the alanine dipeptide (2-state), Lennard-Jones (LJ), and ALDP-S systems?\n\nWe included comprehensive details about each system in Appendix G.1, including state definitions and sample collection procedures. Could you elaborate on which specific details are missing? We will ensure any additional details you suggest are included in our camera-ready version.\n\n**Clarification on System Differences:**\n\n**Two-state alanine dipeptide (ALDP-T):**\n- We split the Boltzmann distribution for alanine dipeptide in vacuum into two metastable state distributions\n- State A: blue shaded region in Fig 2(a) \n- State B: pink shaded region in Fig 2(a)\n- Physical meaning: Free energy difference when alanine dipeptide transitions between metastable conformational states\n\n**ALDP-S (Solvation):**\n- State A: entire Boltzmann distribution for alanine dipeptide in vacuum\n- State B: entire Boltzmann distribution for alanine dipeptide in implicit solvent\n- Physical meaning: Solvation free energy difference between vacuum and implicit solvent environments\n\n> ### Q3: What are the limitations regarding the applicability of the FEAT method to explicit solvent systems or larger (e.g., biomolecular) systems?\n\nExplicit solvent has been a long-standing challenge for simulating biomolecules with machine learning forcefields as explicit solvent molecules are often too large to fit into GPU memory. It is a challenge of independent interest and is orthongoal to what we proposed in our framework. We hence only consider the implicit solvent in our study."}, {"review_id": "B9EdGrEKnT", "reviewer": "Reviewer_7RQk", "summary": "The paper introduces Free Energy Estimators with Adaptive Transport (FEAT), a method for estimating free energy. It requires only samples from the endpoint distributions and learns a continuous transport map between two endpoint distributions, $S_A$​ and $S_B$​, using stochastic interpolants. Bi-directional trajectories that carry samples between $S_A$​ and $S_B$​ are simulated using time-dependent drift and gradient-energy models, parameterized as neural networks. These samples are then used to estimate the free energy. From my perspective, the key novelty is that free energy is not estimated via importance sampling, but instead through a minimum-variance estimator, such as the Bennett Acceptance Ratio (BAR) method. In experiments involving Gaussian Mixture Models, Lennard-Jones particles, and alanine dipeptide systems, FEAT achieves results that are closer to reference values than baseline methods based on target free-energy perturbation approach using the flow-matching and thermodynamic integration.", "questions": "Questions:\n1. This work assumes that the density is the Boltzmann weight $\\exp[-u(x)]$. Can FEAT be generalized to non-Boltzmann sampling?\n2. Theoretical assumptions. When deriving eq. 36, do you assume perfect control, such that samples follow $u_t$ precisely? As I understand it, you assume that forward and backward models are close. What happens if they diverge, e.g., due to numerical errors?\n3. It is unclear how you estimate $\\dot{I}_t$ for learning $v_t$ (eq. 19). I assume the estimate could be noisy. Did you apply any smoothing techniques? If so, which ones?\n4. In Figure 2 (b) and (c), the convergence rate of FEAT is compared with estimates using only forward or backward simulations (eq. 27). To me, it appears that averaging the red curves (eq. 27) would yield a convergence rate similar to the blue curve (eq. 30), which would make for a fairer comparison. Also, the equation numbers in the legends of subplots (b) and (c) seem incorrect.\n5. There is inconsistent notation between the main text and Algorithm 1. In the text, $v$ denotes the vector field, while in Algorithm 1 it is denoted as $u$.\n6. In the Limitations, it is mentioned that FEAT uses only samples from the endpoints. However, if intermediate distributions between the endpoints are available, can FEAT make use of them?\n7. Regarding the training of neural networks: how do you control overfitting? What is the train/validation split ratio?\n8. For completeness, it would be helpful to add in Table 2 the estimated time required for reference value computation. This would highlight the cost of sampling from intermediate distributions and using MBAR.\n9. Typo: Line 185: Corollary 3.1 (Escorted Jarzynski **Equality** ...\n10. How does the number of discretization steps affect FEAT and the baselines? Is FEAT more robust than the baselines in this regard? How does the number of steps required for good performance scale with system size? How does it depend on the similarity between endpoint distributions (e.g., as measured by KL divergence)?\n11. Similar to Question 10, but regarding sample size instead of discretization steps.\n12. Regarding Equations 21 and 22: You use the target score matching approach from [2]. I reviewed that paper, and the choice of intervals $\\mathcal{U}(0, 0.5)$ and $\\mathcal{U}(0.5, 1)$ appears arbitrary. You report results for the same intervals — did you try other options, such as $[0, 0.25]$ and $[0.75, 1]$? Alternatively, could you provide a rationale for the chosen intervals?\n13. Question about generalization: After training, can FEAT be applied to different systems without changing the neural network architecture or retraining? If yes, results from such experiments would be valuable.\n14. What is the performance degradation when omitting the data pre-processing step (canonicalization), both for FEAT and the baselines?\n\n\n[2] B. Máté, F. Fleuret, and T. Bereau. Solvation free energies from neural thermodynamic integration. arXiv preprint arXiv:2410.15815, 2024b\n\n\nI expect the authors to address Questions 1–9 in the rebuttal and to at least partially address some of the experiment-related questions (10–14).", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The proposed approach is theoretically and technically sound. Limitations such as the requirement for access to samples from both systems are clearly stated by the authors.\n2. Overall, the manuscript is well organized and easy to follow. In the appendix, the authors present an extended review of approaches to the problem of free-energy estimation, theoretical proofs, hyperparameters for FEAT and the baselines, as well as a comparison of inference time between FEAT and the baselines.\n3. FEAT outperforms the baselines on the considered datasets, demonstrating its effectiveness in settings where samples from endpoint distributions are available. It is also computationally efficient compared to a baseline that achieves comparable quality.\n4. The theoretical framework is based on the Escorted Jarzynski Equality, Crooks theorem, and Bennett acceptance ratio, and generalizes them to a novel use case involving learned non-equilibrium transport.\n\nWeaknesses:\n1. As there are no well-established state-of-the-art approaches or benchmarks for this task, the value of the work could be improved by evaluating additional baselines (e.g., those mentioned in related works, including DeepBar [1]) and by using datasets from those studies.\n2. The paper could benefit from a more in-depth investigation into the robustness of FEAT (see questions for details). This would help practitioners understand where the method is most applicable.\n3. The limitations section and the algorithm description contain essential information and should be moved from the appendix to the main part of the paper.\n4. There are some typos and inconsistencies in notation throughout the text. Additionally, a few technical details remain unclear (see questions).\n\n\n[1] Ding,  Xinqiang and Zhang,  Bin. DeepBAR: A Fast and Exact Method for Binding Free Energy Computation. The Journal of Physical Chemistry Letters, 2021.", "was_revised": true, "final_justification": "The authors have addressed many of my theoretical and practical concerns, which were listed in the Weaknesses and Questions sections. However, the issue of selecting appropriate time intervals for target score matching remains underexplored, even if the default values \"just work.\" The authors argue that this topic is tangential to their main contribution, but it is, in fact, an essential part of the proposed approach.\nOverall, I am positive about the paper and the authors’ responses, so I have raised my score.", "rebuttal": "Thank you for your constructive review. We are pleased that you found our work theoretically sound, well-organized, effective, and novel. We have addressed your concerns with further experiments and clarifications. We are committed to addressing any further concerns you may have. Should you find our reply satisfactory, we kindly invite you to reconsider your rating.\n## Responses to Weaknesses\n\n> ### W1: Comparison with DeepBAR as an additional baseline\n\nThank you for this excellent suggestion!\nHowever, **FEAT is orthogonal to DeepBAR rather than directly comparable.**\n\nDeepBAR estimates free energy differences by first estimating absolute free energies for each system using generative models (e.g., NF or CNF), then taking their difference. **FEAT can actually improve DeepBAR by providing better transport learning for the absolute free energies.**\n\nWe provide results on GMM and ALDP-S using DeepBAR with CNF and FEAT below:\n\n|| GMM-40D (GT: 0) | GMM-100D (GT: 0) | ALDP-S (reference value: 29.43) |\n|--|--|--|--|\n| DeepBAR-CNF  | -0.06  | 0.16  | 28.68 |\n| DeepBAR-FEAT | -0.02  | -0.20  | 29.38  |\n\nDeepBAR-FEAT outperforms DeepBAR-CNF, especially on ALDP-S. Interestingly, on GMM-100, both achieve better performances compared to standard Target FEP w. FM and FEAT, likely because learning transport from Gaussian to GMM is easier than directly learning between two complex GMMs.\n\nWe will include this discussion and comparison in our camera-ready version.\n\n> ###  W2: Additional baselines from related works\n\nSince DeepBAR did not release their datasets, we evaluated on a comparable system: Chignolin (175 atoms, 525 dimensions) in vacuum vs. implicit solvent:\n- Reference value (MBAR): 190.887 kcal/mol\n- DeepBAR-FEAT result: **190.784  +- 0.414 kcal/mol** (due to time limit, we did not train 3 times, and this std is obtained by doing 3 inferences with one model.)\n\nThis showcased our approach's scalability.\n\n## Responses to Questions \n\n> ### Q1: Can FEAT be generalized to non-Boltzmann sampling?\n\nYes. FEAT can handle any distribution admitting a differentiable density $p(x)$. The Boltzmann form is not required: our framework applies to any probability distribution satisfying these mild regularity conditions.\n\n> ### Q2: When deriving Eq. 36, do you assume perfect control?\n\nNo. Equation 36 relies entirely on importance sampling (lines 619-624) and does not require perfect control. This is a key advantage of FEAT: intermediate samples need not remain at equilibrium.\n\nHowever, we do need the control to be \"reasonable\" with sufficient overlap between forward and backward processes. Poor control can increase estimation error, but this is easily detected through:\n- Quality of transported samples\n- Gap between ELBO and EUBO \n- Discrepancy between forward and backward estimators\n\n> ### Q3: How do you estimate $\\dot I_t$  for learning $v_t$?\n\nThis appears to be a misunderstanding. The derivative of the interpolant $\\dot I_t$ can be calculated exactly from our Eq.17: it does not need to be estimated or learned. Instead, we use it to learn $v_t(x) = \\mathbb{E}[\\dot I_t| I_t = x]$ (Eq.18; Eq. 2.10 in [1] ), which is exactly the minimizer of the loss in Eq.19 and is not noisy.\n\n[1] Michael S. Albergo, et al. Stochastic Interpolants: A Unifying Framework for Flows and Diffusions\n\n> ### Q4: Comparison between averaged forward-backward estimators vs. min-variance estimator\n\nThis is a great suggestion! In fact, **the minimum-variance estimator (blue curve) is theoretically guaranteed to perform no worse than averaged forward-backward estimators (red curves)**, as it's derived to minimize estimation error.\n\nTo see this, we provide an \n**empirical verification on ALDP-S:**\n- After 2000 iters: Blue = 29.30 +- 0.36; Avg red = 28.86 +- 1.19\n- After 5000 iters: Blue = 29.54 +- 0.16; Avg red = 29.56 +- 0.73\n\n> ### Q6: Can FEAT utilize intermediate distributions if available?\n\nYes. We can learn multiple FEATs in sequence: Given intermediate distributions $U_2...U_{N-1}$ between $A$ (energy $U_1$) and $B$ (energy $U_N$), we learn FEAT between each neighboring pair $U_i$ and $U_{i+1}$. The total $\\Delta F$ is the sum of individual estimates. This approach may reduce individual network burden compared to learning a single $A\\to B$ transport.\n\n> ###  Q7: How do you control overfitting? \n\nWe answer this from two perspectives:\n\n**a) Train/test split:**\nFor a given pair of state A & B, we aim to fit a transport between A and B as much as possible, and in theory we assume that we can sample sufficient configurations in both states. From this perspective, our approach is more akin to stochastic approximation methods from Robbins-Monro [2] and therefore has minimal overfitting issues. \n\nOn the other hand, when the data set is provided beforehand and kept fixed, it may not be representative enough, and using the same set for training and testing may introduce bias. To this end, we provide an 8:2 train/test split result on ALDP.  The result is not significantly better than that reported in our manuscript, indicating that the dataset we used is sufficient.\n\n| Target | Result | Ground Truth |\n|----|---|-----|\n| ALDP-S | 29.45 ± 0.06 | 29.43 ± 0.01 |\n| ALDP-T | -4.39 ± 0.09 | -4.25 ± 0.05 |\n\n**b) Hyperparam tuning:**\nAnother question is whether a train/valid split is needed for hyperparam tuning. \nThe answer is No:\nin our approach, we tune all hyperparameters based on transport quality by monitoring the energy of transported training samples, or the gap between forward and backward estimators. These validation metrics eliminate the need for separate validation sets.\n\n[2] Robbins, H. and Monro, S., 1951. A stochastic approximation method. The annals of mathematical statistics, pp.400-407.\n\n> ### Q8: Reference computation times for Tab 2 to highlight the cost of sampling from intermediate distributions\n\nFor ALDP, generating samples for each target requires ~1 day on our machine. Collecting all the targets for MBAR can hence take between 1-10 days, depending on whether the simulations are run in parallel.\n\nIn contrast, our training process takes only 1-2 hours, which is significantly faster than sampling from all intermediate densities.\nWe will include these rough numbers and the corresponding discussion in our camera-ready version.\n\n> ###  Q10 & Q11: Robustness w.r.t. discretization steps and sample size\n\nFollowing your suggestion, we conducted robustness analysis on GMM:\n\n**FEAT**\n| #step| sample size| GMM-40D | GMM-100D |\n|-|-|--|--|\n| 50 | 5k| -0.05 +- 0.33 | -4.04 +- 1.92  |\n| 100|5k| 0.12 +- 0.07  | -6.43 +- 1.72  |\n| 500 |5k| 0.00 +- 0.06  | -3.59 +- 1.06  |\n| | | |  |\n| 500|500  | 0.13 +- 0.09  | -5.56 +- 1.87  |\n| 500 |1k | -0.04 +- 0.05 | -6.46 +- 1.86  |\n| 500 |5k| 0.00 +- 0.06  | -3.59 +- 1.06  |\n\n**Target FEP**\n| #step| sample size| GMM-40D | GMM-100D |\n|-|-|--|--|\n| 50 | 5k| -0.98 +- 0.23 | -15.44 +- 6.02 |\n| 100|5k| -0.48 +- 0.17 | -12.10 +- 4.80 |\n| 500 |5k|-0.12 +- 0.08 | -13.72 +- 2.46 |\n| | | | |\n| 500|500  |  -0.16 +- 0.37 | -22.24 +- 3.43 |\n| 500 |1k |  0.02 +- 0.18  | -19.73 +- 2.13 |\n| 500 |5k|-0.12 +- 0.08 | -13.72 +- 2.46 |\n\nThese results **highlight FEAT's greater robustness toward fewer steps/samples**, compared to Target FEP w flow matching. The results obtained with fewer steps (larger discretisation error) also reflect our discussion in lines 238-239.\n\n> ###  Q12: Target score matching with different time intervals\n\nFollowing your suggestions, we evaluate FEAT with $t \\sim U(0, 0.25)$ and $U(0.75, 1)$ in TSM on ALDP-S.  It achieves $29.37 \\pm 0.02$ across 3 independent runs. \nCompared to the choice with $U(0, 0.5)$ and $U(0.5, 1)$ ($29.38\\pm 0.04$), we do not see obvious difference.\n\nThis is because TSM helps the score learn better at boundary.  As long as we sample $t$ close to the boundary, TSM can provide useful signal. \nAlso, different from neural TI that fully relies on the learned energy, FEAT rely on non-equilibrium transports with both the score and the vector field, leading to a higher robustness towards the TSM choice.\n\n> ###  Q13: Can FEAT be applied to different systems without retraining?\n\nYes. If we train the network using datasets from multiple targets and condition the network on the target parameters, the resulting model becomes transferable, allowing us to apply FEAT to similar systems without retraining.\n\nWe showcase this on GMM-40 with different scaling factor: concretely, we scale the state B with scaling $0.5, 0.7, 0.9, 1.1, 1.3, 1.5$ when training. Our network will also take the scaling as an input. After training, we evaluate $\\Delta F$ on unseen scaling factors and report the error and std across 3 runs in the following table:\n\n| Scaling | 0.45 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.55 |\n|--|--|--|--|--|--|--|--|\n| Error +- std  | -3.52 +- 1.13 | -0.07 +- 0.05 | -0.02 +- 0.06 | -0.01 +- 0.06 | 0.01 +- 0.04 | 0.003 +- 0.04 | 0.03 +- 0.13 |\n\nThese results show that the transferable FEAT model achieves good accuracy across a range of unseen targets. Also, as expected, interpolation yields better performance than extrapolation.\n\n> ### Q14: Performance without data preprocessing\n\nOmitting the data processing step significantly complicates training, resulting in larger bias and variance. Following your suggestion, we provide an ablation study on ALDP-S.  Without pre-processing:\n- FEAT: $37.40 \\pm 0.38$\n- Target FEP w FM: $44.2 \\pm 4.0$. \n \nNeither method attains a reasonable value, while FEAT exhibits a slightly better performance.\n\nHowever, we emphasize that this is not a weakness of our proposed framework. Rather, it reflects a general challenge in generative modeling. We also expect that using higher-capacity architectures (e.g., transformers) could help mitigate this issue. \n\n## Formatting and typos (Weakness 3/4 and Q5/9):\n\nThank you for pointing these typos out! We will fix them in our camera-ready version. \n\nBroader Impact is not counted to the page limit per instruction, so our main text is within 9 pages."}, {"review_id": "ZXZuhyTr9b", "reviewer": "Reviewer_Enj8", "summary": "This work presents FEAT, a novel framework for free energy calculation that provides consistent, minimum-variance estimators based on escorted Jarzynski equality and controlled Crooks theorem. The framework leverages stochastic interpolant to learn transports from two endpoint systems directly, with no necessity for intermediate states. It has been proved that the estimator is asympototically unbiased even with imperfect learned potentials and discretization errors. Experiments on Gaussian mixtures, LJ and alanine-dipeptide show promising improvements on free energy estimations compared to all baselines.", "questions": "1. From my perspective, since the proposed method relies entirely on neural networks and potential energy calculations for free energy estimation without requiring MD simulations, it should be applicable to larger molecular systems in principle. Could the authors provide experimental results on more complex molecules like proteins, along with an analysis of the computational efficiency?\n2. When training the stochastic interpolant, each iteration requires selecting a data pair $(x_a,x_b)$. In the case of FEAT, are the data points independently sampled from $\\mu_a$ and $\\mu_b$ during training, or are the datasets $x_a^{1:N}$ and $x_b^{1:N}$ pre-paired in a one-to-one fashion? A brief explanation of whether these two approaches would lead to different training outcomes would be appreciated.\n3. There appears to be a typo in Figures 2(b) and 2(c): the references should be to Eq. (27) and Eq. (30), rather than Eq. (24)/(25) and Eq. (27).", "limitations": "It is strongly recommended that FEAT be evaluated on larger molecular systems to demonstrate its applicability to more complex, real-world scenarios.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n- The paper is exceptionally well-organized and clearly articulated, making it easy to follow even for readers without a background in computational physics.\n- The proposed method demonstrates strong originality and practical relevance, and is supported by rigorous theoretical justifications.\n- The proposed method enables the estimation of free energy without requiring access to intermediate states, and it is theoretically shown to remain a consistent estimator even when the learned score is imperfect on the boundaries. This significantly reduces the reliance on computational resources and model parametrization.\n\n**Weaknesses**\n- The proposed method estimates free energy without relying on MD simulations and is, in principle, fully extendable to macromolecular systems. However, it is regrettable that the experiments are limited to small molecules (e.g., alanine dipeptide), without demonstrating its applicability to larger systems.", "was_revised": true, "final_justification": "My main concern was whether FEAT can scale to larger molecular systems. In the rebuttal, the authors provided additional experiments on alanine dipeptide and Chignolin, where FEAT demonstrated promising results. I believe this supports the practical applicability of FEAT to larger systems. Therefore, I will maintain my score.", "rebuttal": "Thank you for your detailed and insightful review. We are delighted that you found our work exceptionally well-organized, original, effective, and significant. We now address your concerns with new experiments on larger molecular systems and clarifications on experimental details:\n\n## Responses to Weaknesses and Questions\n\n> ### Q1 & Weakness: Experimental results on more complex molecules \n\nThank you for your great suggestions. Following your recommendation, we now provide results on significantly larger systems:\n\n**Alanine Tetrapeptide Results:**\n- System size: 43 atoms (129 dimensions)\n- FEAT estimate: **65.51 ± 1.51 kcal/mol**\n- Reference value (MBAR with 20 intermediate distributions): **64.09 kcal/mol**\n\n**Chignolin Results:**\n- System size: 175 atoms (525 dimensions)\n- For this larger system, we found that it is more effective to decompose the estimation task: instead of directly learning a transport map between the vacuum and solvent states, we estimate the free energy using FEAT in two stages—from a Gaussian to vacuum, and from a Gaussian to solvent—and then take the difference between the two estimates. \n- FEAT estimate: **190.784  +- 0.414 kcal/mol** (due to time limit, we did not train 3 times, and this std is obtained by doing 3 inferences with one model.)\n- Reference value (MBAR with 20 intermediate distributions): **190.887 kcal/mol**\n\nThese results demonstrate FEAT's applicability to larger, more realistic systems. We will update these results in our camera-ready version.\n\n> ### Q2: Data pair in stochastic interpolant training\n\nThis is an excellent question.\nThe FEAT framework is general and supports multiple pairing strategies, for example:\n- OT: Using optimal transport (OT) plans for structured correspondence\n- Random: Independent sampling from each endpoint distribution\n\nIn our experiments on simpler systems such as GMM-40 and LJ-55, we use randomly selected pairs. For more complex targets, we sample batches randomly but apply mini-batch OT to determine better pairings, which helps to stabilize training and improve network fitting. We found that omitting the OT step leads to slower convergence and reduced performance, as random pairing can lead to less \"straight\" vector fields, making training more challenging.\n\nIn table 3, we have a hyperparameter called \"OT pair\" to reflect if we use this approach in our training. We will provide a brief discription and discussion in our camera-ready version.\n\n\n> ### Q3: Figure 2 equation reference typo\n\nThank you for catching it. We will fix in our camera-ready version."}, {"review_id": "vecnqTrCrD", "reviewer": "Reviewer_HXX2", "summary": "FEAT introduces a learned stochastic‐dynamics framework for free‐energy estimation that adapts transport maps via time‑dependent vector fields and score functions. It unifies Jarzynski’s equality and Bennett’s acceptance‐ratio into two asymptotically unbiased, minimum‑variance estimators (FEAT‑J and FEAT‑BAR). By training these flows to minimize dissipation, FEAT reduces estimator variance without requiring intermediate sampling.", "questions": "In Table 3:\nRow about parameter size: The network sizes are listed as \"5, 400\" and \"4, 64.\"\nWhat exactly do these numbers refer to? Are they the number of layers and hidden units, or do they represent the total number of learnable parameters?\nIf they do correspond to a small number of parameters, can you justify the use of an H100 GPU?\n\nThroughout the dataset description, I did not find any information about train/test splitting.\nCould you clarify how the data was handled during training and validation?\nWas any explicit split used, or is this more of an overfitting-style exercise without separate validation?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength\n\nStrong theoretical foundation, supported by extensive proofs.\n\nWeaknesses\n\nIn Table 1 results for Neural TI perform poorly: although the text offers some justification, something should be done to make these results a fair comparison, otherwise these results do not have any significant meaning.\n\nFor certain systems (specifically ALDP‑S and ALDP‑T), the standard deviations are so large that comparing only the means does not ensure these estimates are trustworthy or reproducible under slightly different setups.\n\nThe lack of available code enhances these concerns, as it prevents independent verification and deeper investigation into the above issues.", "was_revised": true, "final_justification": "After the rebuttal phase, the authors addressed my concerns by changing how some experiments were conducted. With the new results, I feel more inclined to accept the paper", "rebuttal": "Thank you for your review. We have addressed your concerns with improved baseline comparisons, detailed statistical analysis, experimental details, and code availability. We are happy to answer any further questions you might have. Should you find our reply satisfactory, we kindly invite you to reconsider your rating.\n\n## Responses to Weaknesses\n\n> ### 1. In Table 1 results for Neural TI perform poorly: although the text offers some justification, something should be done to make these results a fair comparison, otherwise these results do not have any significant meaning.\n\nThank you for your suggestion. In our experiment, we tried to keep the network architecture as consistent as possible across different approaches. However, we fully agree that this is not ideal for neural TI, which benefits from task-specific architectural design to ensure boundary conditions and also mitigate the \"blindless issue\" of score matching losses (e.g., score is insensitive to error in mode weights when modes are separate).\n\nFollowing your suggestion, we **re-evaluated neural TI on GMM-40 and LJ using a better-designed network architecture** in the following.\n\n#### **Updated network design for neural TI:**\n\n- for GMM, we set the energy network to be \n\n$\nU(x_t, t) =  a_t U_A(x_t) + b_t U_B(x_t) + c_t U_\\theta (x_t, t)\n$\n\nwhere $a_t = \\exp[f_\\theta(t) - f_\\theta(0)] \\cdot (1-t)$, $b_t = \\exp[g_\\theta(1) - g_\\theta(t)] \\cdot t$ and $c_t = \\exp(h_\\theta(t))  \\cdot t \\cdot (1-t)$, where $f_\\theta, g_\\theta, h_\\theta, U_\\theta$ are neural nets and by which we can exactly ensure the boundary condition.\n\n- for LJ, $U_A$ and $U_B$ are more sensitive to noisy $x_t$. Therefore, inspired by [1],  we set the energy network to be \n\n$\nU(x_t, t) = b_t \\cdot U_\\text{LJ}(x_t, r_t, a_t) + c_t \\cdot U_\\theta(x_t, t)\n$\n\nwhere $r_t$ is the radius parameter in LJ, ranging from  0 to 1. $a_t = \\exp(\\alpha_\\theta(t) )\\cdot t \\cdot (1-t)$ is a smooth parameter inspired by [1]. $b_t = 1 - \\exp(\\beta_\\theta(t))\\cdot t\\cdot (1-t)$ and $c_t = \\exp(\\gamma_\\theta(t)) \\cdot t \\cdot (1-t)$ are scalar to ensure boundary conditions.\n\n\n#### **Updated neural TI results:**\n\nWith these updated architectures, we obtained the following **updated results for neural TI**:\n\n- **GMM-40: 0.366** (reference value: 0)\n- **LJ-55: 235.894** (reference value: 234.77)\n- **LJ-79: 354.8852** (vs. reference 357.43)\n\nThese results show that neural TI can achieve competitive performance when supported by carefully designed network architectures.\nWe will provide updated results on other targets and provide error bars, and we will ensure a fairer discussion on neural TI  in our final version.\n\n[1] Máté, B., Fleuret, F. and Bereau, T., 2025. Solvation free energies from neural thermodynamic integration. The Journal of Chemical Physics, 162(12).\n\n\n> ### 2. For certain systems (specifically ALDP), the standard deviations are so large that comparing only the means does not ensure these estimates are trustworthy or reproducible.\n\nWe completely agree that both mean and variance are crucial for assessing method reliability. We have conducted extensive additional experiments to address this concern.\n\nWe performed **12 runs** (6 different model seeds × 2 estimates per model) and provide more statistics:\n\n| Method | All values                                                                                      | Mean        | Std Dev     |\n|--------|-------------|-------------|-------------|\n| FEAT   | 29.43451, 29.49945, 29.37685, 29.40085, 29.32738, 29.37606, 29.50049, 29.44383, 29.34062, 29.32738, 29.50566, 29.43218 | 29.41 | 0.06 |\n| Target FEP   | 29.26557, 29.68326, 29.16319, 29.59889, 29.36639, 29.50764, 29.59858, 29.52172, 29.34441, 29.47849, 29.35864, 29.34831 | 29.44  | 0.15 |\n\nTheir 25, 50 and 75-quantiles are:\n\n- FEAT: 29.367, 29.417, 29.458\n- Target FEP: 29.347, 29.422, 29.541\n\nWe can observe that FEAT gives more stable results when taking standard deviation and quantiles into account.\n\n> ### 3. Code availability \n\nWe totally understand your concern. Due to this year’s policy, we are not allowed to directly provide you with links pointing to our code; therefore, **we have sent our code link to AC**.  \n\nAdditionally, we will make our code publicly available in the next version of our paper.\n\n## Responses to Questions\n\n> ### Q1: Network architecture specifications in Table 3\n\nSorry for the confusion. \"5, 400\" refers to 5 layers of neural networks with 400 hidden dimensions; similarly, \"4, 64\" refers to 4 layers of neural networks with 64 hidden dimensions. \n\nOur method does not necessarily require an H100 to train, smaller GPUs also suffice.  We further report the GPU usage of our targets:\n\n| target | Batch size | GPU usage |\n|---|---|---|\n| LJ-55 | 100 | ~28 G |\n| LJ-79 | 30 | ~18 G |\n| LJ-128 | 20 | ~19 G |\n| GMM-40 | 1000 | ~1G |\n| GMM-100 | 1000 | ~2G |\n| ALDP | 500 | ~20G |\n\n> ### Q2: Train/test splitting and validation procedures\n\n\n\nWe address this from two perspectives:\n\n**a) Train/test split:**\nFor a given pair of states A & B, we aim to fit a transport between A and B as much as possible, and in theory, we assume that we can sample sufficient configurations in these states.\nFrom this perspective, our approach is more akin to stochastic approximation methods from Robbins-Monro [2] and therefore has minimal overfitting issues. \n\nOn the other hand, when the data set is provided beforehand and kept fixed, it may not be representative enough, and using the same set for training and testing may introduce bias.\nTo this end, we provide an 8:2 train/test split result on ALDP. \nThe result is not significantly better than that reported in our manuscript, indicating that the dataset we used is sufficient.\n\n| Target | Result | Ground Truth |\n|--------|--------|--------------|\n| ALDP-S | 29.45 ± 0.06 | 29.43 ± 0.01 |\n| ALDP-T | -4.39 ± 0.09 | -4.25 ± 0.05 |\n\n**b) Hyperparameter tuning:**\nAnother question is whether a train/validation split is needed for hyperparameter tuning. \nThe answer is No: in our approach, we tune all hyperparameters based on transport quality by monitoring the energy of transported training samples and the gap between forward and backwards estimators.\n\nThese validation metrics eliminate the need for separate validation sets.\n\n[2] Robbins, H. and Monro, S., 1951. A stochastic approximation method. The annals of mathematical statistics, pp.400-407."}]

Author rebuttal:

Meta-review: The FEAT framework introduced in this paper estimates free energy using adaptive transport. It is well-grounded in theory, and the paper is well written, and demonstrates improvements over baselines. The new results added during rebuttal were well taken by the reviewers, and all of them recommend acceptance. I also vote to accept the paper as FEAT represents a timely contribution to AI for science.
Decision: Accept (poster)