Equivariant Neural Diffusion for Molecule Generation

Authors: françois  cornet
grigory  bartosh
mikkel n schmidt
christian a naesseth
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 17:12:07

Abstract: We introduce Equivariant Neural Diffusion (END), a novel diffusion model for molecule generation in 3D that is equivariant to Euclidean transformations. Compared to current state-of-the-art equivariant diffusion models, the key innovation in END lies in its learnable forward process for enhanced generative modelling. Rather than pre-specified, the forward process is parameterized through a time- and data-dependent transformation that is equivariant to rigid transformations.  Through a series of experiments on standard molecule generation benchmarks, we demonstrate the competitive performance of END compared to several strong baselines for both unconditional and conditional generation.

Main paper: 
 introduction: 
 The discovery of novel chemical compounds with relevant properties is critical to a number of scientific fields, such as drug discovery and materials design (Merchant et al., 2023) . However, due to the large size and complex structure of the chemical space (Ruddigkeit et al., 2012) , which combines continuous and discrete features, it is notably difficult to search. Additionally, ab-initio Quantum Mechanics (QM) methods for computing target properties are often computationally expensive, preventing brute-force enumeration. While some of these heavy computations can be amortized through learned surrogates, the need for innovative search methods remains, and generative models have recently emerged as a promising avenue (Anstine and Isayev, 2023) . Such models can learn complex data distributions, that, in turn, can be sampled from to obtain novel samples similar to the original data. Compared to other data modalities such as images or text, molecules present additional challenges as they have to adhere to strict chemical rules, and obey the symmetries of the 3D space.Currently, the most promising directions for molecule generation in 3D are either auto-regressive models (Gebauer et al., 2019; 2022; Luo and Ji, 2022; Daigavane et al., 2024) building molecules one atom at a time, or Diffusion Models (DMs) (Hoogeboom et al., 2022; Vignac et al., 2023; Le et al., 2024) that learn to revert a corruption mechanism that transforms the data distribution into noise. As both approaches directly operate in 3D space, they can leverage architectures designed for machine learned force fields (Unke et al., 2021) , that were carefully developed to encode the symmetries inherent to the data (Schütt et al., 2017; 2021; Batzner et al., 2022; Batatia et al., 2022) .The success of DMs has not been limited to molecule generation, and promising results have been demonstrated on a variety of other data modalities (Yang et al., 2023) . Nevertheless, most existing DMs pre-specify the forward process, forcing the reverse process to comply with it. A recent line of work has sought to overcome that limitation and improve generation by replacing the fixed forward process with a learnable one (Bartosh et al., 2023; Nielsen et al., 2024; Bartosh et al., 2024) .Contributions In this paper, we present Equivariant Neural Diffusion (END), a novel diffusion model for molecule generation in 3D that (1) is equivariant to Euclidean transformations, and (2) features a learnable forward process. We demonstrate competitive unconditional molecule generation performance on the QM9 and GEOM-Drugs benchmarks. For conditional generation driven by composition and substructure constraints, our approach exhibits a substantial performance gain compared to existing equivariant diffusion models. Our set of experiments underscores the benefit of a learnable forward process for improved unconditional and conditional molecule generation.In this section, we omit the invariant features, and consider that x, z t , ε ∈ R M ×d are all collections of vectors.In opposition to the notations introduced in Section 3, we would like F φ and xθ (which ultimately are neural networks) to operate in ambient space directly, i.e. on z t , x, ε ∈ R with R = {v ∈ R M ×d :1 M M i=1 v i = 0}, instead of R (M -1)×d as initially presented. To do so, we use the results from Garcia Satorras et al. ( 2021) that showed that the Jacobian of the transformation F φ can be computed in ambient space, for z t and ε that live in the linear subspace R, provided that the transformation F φ (invertible with respect to ε) leaves the center of mass of ε unchanged.Considering flat representations of z t , x, ε ∈ R M •d , the parameterization of F φ introduced in Eqs. ( 6), ( 8) and ( 9) can be adapted as follows to achieve such propertyEQUATIONEQUATIONEQUATIONwhereT ∈ R M •d×d = [I d , I d , ...] ⊤ , and 1 M T T ⊤ corresponds to the linear operator computing the center of mass. In Eq. ( 11), the unconstrained mean output μφ (x, t) is simply projected onto the 0-CoM subspace, thereby inducing no translation. In Eq. ( 12), the unconstrained output of Ũφ (x, t) is first projected onto the 0-CoM subspace, before being translated back to the initial center of mass.With this adapted formulation, we need to be able to (1) compute the latent variable z t given ε, (2) evaluate the Jacobian of the transformation F φ , and (3) evaluate the inverse transformation F -1 φ .Computing z t from ε Given that ε ∈ R, obtaining z t ∈ R from ε simply amounts to (1) computing zt = μφ (x, t) + Ũφ (x, t)ε, and then (2) removing the center of mass from zt -the second term in Eq. ( 12) induces no translation as ε ∈ R.Computing |J Fφ | and F -1 φIn what follows, we shorten the notation, and denote U φ (x, t) by U and Ũφ (x, t) by Ũ . To leverage known identities, we start by reorganizing Eq. ( 12), asEQUATIONThe Jacobian of F φ is given by the determinant of U , and the latter can be derived by leveraging the Matrix Determinant Lemma,EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONwhere Ũ m denotes the m-th d × d-block in Ũ , andV = 1 M M m=1 ( Ũ m ) -1 is a d × d-matrix.Computing the Jacobian therefore amounts to compute the inverse of M d × d-matrices and the determinant of M + 1 d × d-matrices. In practice, d = 3 such that all computations can be performed in closed-form.Regarding F -1 φ , we do not need to evaluate the inverse transformation itself, but instead evaluate ε given z t ,ε = U -1 z t -µ(x, t) .The inverse U -1 can be obtained via the Woodbury matrix identity,U -1 = Ũ + 1 M T T ⊤ (I M •d -Ũ ) -1 , = Ũ -1 - 1 M Ũ -1 T V -1 T ⊤ ( Ũ -1 -I M •d ), = Ũ -1 (I -C), where V = 1 M M m=1 ( Ũ m ) -1as previously defined in Eq. ( 18), andC = 1 M T V -1 T ⊤ ( Ũ -1 -I M •d ).Given the specific structure of C, the computation of ε can be simplified toε = U -1 zt , = Ũ -1 (I -C) zt , = Ũ -1 zt -c ,where z = z t -µ(x, t) and c = C zt acts as a translation operator. We note that computing the inverse transformation requires to invert M + 1 d × d-matrices, but as d = 3 in practice, all computations can be performed in closed-form. 
background: 
 We begin by establishing the necessary background for generative modeling of geometric graphs. We first introduce the data representation and its inherent symmetries. We then discuss Diffusion Models (DMs), and more specifically the Equivariant Diffusion Model (EDM) (Hoogeboom et al., 2022) . Finally, we present the Neural Flow Diffusion Models (NFDM) framework (Bartosh et al., 2024) .Molecules as geometric graphs in E(3) We consider geometric graphs embedded in 3-dimensional Euclidean space that represent molecules. Formally, each atomistic system can be described by a tuple x = (r, h), where r = (r 1 , ..., r M ) ∈ R M ×3 form a collection of vectors in 3D representing the coordinates of the atoms, and h = (h 1 , ..., h M ) ∈ R M ×D are the associated scalar features (e.g. atomic types or charges). When dealing with molecules, we are particularly interested in E(3), the Euclidean group in 3 dimensions, generated by translations, rotations and reflections. Each group element in E(3) can be represented as a combination of a translation vector t ∈ R 3 and an orthogonal matrix R ∈ O(3) encoding rotation or reflection. While scalar features h remain invariant, coordinates r transform under translation, rotation and reflection as Rr+t = (Rr 1 +t, ..., Rr M +t).Equivariant functions A function f : X → Y is said to be equivariant to the action of a group G, orG-equivariant, if g • f (x) = f (g • x), ∀g ∈ G. It is said to be G-invariant, if f (x) = f (g • x), ∀g ∈ G.In the case of a function f : (R M ×3 × R M ×D ) → (R M ×3 × R M ×D ) operating on geometric graphs, the function is said to be E(3)-equivariant if Ry (r) + t, y (h) = f Rr + t, h , ∀ R ∈ O(3) and t ∈ R 3 , where y (r) and y (h) denote the output related to r and h respectively. There exists a large variety of graph neural network architectures designed to be equivariant to the Euclidean group (Schütt et al., 2017; 2021; Batzner et al., 2022; Batatia et al., 2022) .Equivariant distributions A conditional distribution p(y|x) is equivariant to rotations and reflections when p(y|x) = p(Ry|Rx), ∀ R ∈ O(3), while a distribution is said to be invariant when p(x) = p(Rx), ∀ R ∈ O(3). Regarding translation, it is not possible to have a translation-invariant non-zero distribution, as it would require that p(x) = p(x + t), ∀t ∈ R 3 , x ∈ R M ×3 , which would mean that p(x) cannot integrate to 1 (Garcia Satorras et al., 2021) . However, a translation-invariant distribution can be constructed in the linear subspace R, where the centre of gravity is fixed to 0 (i.e. zero CoM subspace): R = {r ∈ R M ×3 : 1 M M i=1 r i = 0} (Xu et al., 2022) . As R can be shown to be intrinsically equivalent to R (M -1)×3 (Bao et al., 2023) , we will consider in what follows that r is defined in R (M -1)×3 for ease of notation.Diffusion Models (DMs) (Sohl-Dickstein et al., 2015; Ho et al., 2020) are generative models that learn distributions through a hierarchy of latent variables, corresponding to perturbed versions of the data at increasing noise scales. DMs consist of a forward and a reverse (or generative) process. The Equivariant Diffusion Model (EDM) (Hoogeboom et al., 2022) is a particular instance of a DM, where the learned marginal p θ (x) is made invariant to the action of translations, rotations and reflections by construction. Intuitively, this means that the likelihood of a given molecule under the model does not depend on its orientation.The forward process perturbs samples from the data distribution, x ∼ q(x), over time through noise injection, resulting in a trajectory of latent variables (z t ) t∈[0,1] , conditional on x. The conditional distribution for (z t ) t∈[0,1] given x, can be described by an initial distribution q(z 0 |x) and a Stochastic Differential Equation (SDE),d{z (r) t , z (h) t } = f (t) z (r) t , z (h) t dt +g(t) d{w (r) , w (h) },where the drift f (t) and volatility g(t) are scalar functions of time, and w (r) and w (h) are two independent standard Wiener processes defined in R (M -1)×3 and R M ×D respectively. Specifically, EDM implements the Variance-Preserving SDE (VP-SDE) scheme (Song et al., 2020) , with f (t) = -1 2 β(t) and g(t) = β(t) for a fixed schedule β(t). Due to the linearity of the drift term, the conditional marginal distribution is known in closed-form (Särkkä and Solin, 2019) , and can be reconstructed asq [z (r) t , z (h) t ] [r, h] = q(z (r) t |r)q(z (h) t |h) = N (z (r) t , |α t r, σ 2 t I) • N (z (h) t , |α t h, σ 2 t I),whereα t = exp -1 2 t 0 β(s) ds and σ t = 1-exp -1 2 t 0 β(s) ds .It evolves from a low-variance Gaussian centered around the data q(z 0 |x) ≈ N (z 0 |x, δ 2 I) to an uninformative prior distribution (that contains no information about the data distribution), i.e. a unit Gaussian q(z 1 |x) ≈ N (z 1 |0, I).Starting from the prior [z (r) 1 , z (h) 1 ] ∼ N (z (r) t |0, I) • N (z (h)t |0, I), samples from q(x) can be generated by reversing the forward process. This can be done by following the reverse-time SDE (Anderson, 1982) ,dz t = f (t) z (r) t , z (h) t -g 2 (t) ∇ z (r) t log q(z t ), ∇ z (h) t log q(z t ) dt +g(t) d{ w(r) , w(h) },where w(r) and w(h) are independent standard Wiener processes defined in R (M -1)×3 and R M ×D , respectively, with time flowing backwards. DMs approximate the reverse process by learning an approximation of the score function ∇ zt log q(z t ) parameterized by a neural network s θ (z t , t). With the learned score function s θ (z t , t), a sample z 0 ∼ p θ (z 0 ) ≈ q(z 0 ) ≈ q(x) can be obtained by first sampling from the prior [z(r) 1 , z (h) 1 ] ∼ N (z (r) t |0, I) • N (z (h)t |0, I), and then simulating the reverse SDE,dz t = f (t) z (r) t , z (h) t -g 2 (t) s (r) θ (z t , t), s (h) θ (z t , t) dt +g(t) d{ w(r) , w(h) },where the true score function has been replaced by its approximation s θ (z t , t). In EDM, the approximate score is parameterized through an equivariant function:s θ (z t , t) = s (r) θ (z t , t), s (h) θ (z t , t) such that s θ ([Rz (r) t , z (h) t ], t) = Rs (r) θ (z t , t), s (h) θ (z t , t) , ∀ R ∈ O(3). Practically, this is realized through the specific parameterization, s θ (z t , t) = α t xθ (z t , t) -z t σ 2 t ,where the data point predictor xθ is implemented by an equivariant neural network.Optimization The data point predictor xθ , or s θ , is trained by optimizing the denoising score matching loss (Vincent, 2011) ,L DSM = E u(t),q(x,zt) λ(t) s θ (z t , t) -∇ zt log q(z t |x) 2 2 ,where λ(t) is a positive weighting function, and u(t) is a uniform distribution over the interval [0, 1].Neural Flow Diffusion Models (NFDM) (Bartosh et al., 2024) are based on the observation that latent variables in DMs, i.e. z t , are conventionally inferred through a pre-specified transformation -as implied by the chosen type of SDE and the noise schedule. This potentially limits the flexibility of the latent space, and makes the learning of the reverse (generative) process more challenging.In contrast to conventional DMs, NFDM define the forward process implicitly through a learnable transformation F φ (ε, t, x) of injected noise ε, time t, and data point x. The latent variables z t are obtained by transforming noise samples ε, conditional on data point x and time step t: z t = F φ (ε, t, x). If F φ is differentiable with respect to ε and t, and invertible with respect to ε, then, for fixed x and ε, samples from q φ (z t |x) can be obtained by solving the following conditional Ordinary Differential Equation (ODE) until time t,EQUATIONwith z 0 ∼ q(z 0 |x). While F φ and q(ε) define the conditional marginal distribution q φ (z t |x), we need a distribution over the trajectories (z t ) t∈ [0, 1] . NFDM obtain this through the introduction of a conditional SDE starting from z 0 and running forward in time. Given access to the ODE in Eq. ( 1) and the score function ∇ zt log q φ (z t |x), the conditional SDE sharing the same conditional marginal distribution q φ (z t |x) is given byEQUATIONwhere the score function ofq φ (z t |x) is ∇ zt log q φ (z t |x) = ∇ zt log q(ε) + log J -1 F , with ε = F -1 φ (z t , t, x), and J -1 F = ∂F -1 φ (zt,t,x) ∂zt .Reverse (generative) process A conditional reverse SDE that starts from z 1 ∼ q(z 1 ), runs backward in time, and reverses the conditional forward SDE from Eq. ( 2) can be defined asdz t = f B φ (z t , t, x) + g φ (t) d w with f B φ (z t , t, x) = f φ (z t , t, x) - g 2 φ (t) 2 ∇ zt log q φ (z t |x). (3)As x is unknown when generating samples, we can rewrite Eq. ( 3) with the prediction of x instead,EQUATIONwhere xθ is a function that predicts the data point x. Provided that the reconstruction distribution q(z 0 |x) and prior distribution q(z 1 ) are defined, this fully specifies the reverse (generative) process.Optimization The forward and reverse processes can be optimized jointly by matching the drift terms of the true and approximate conditional reverse SDEs through the following objective,EQUATIONwhich can be shown to be equivalent to minimizing the Kullback-Leibler divergence between the posterior distributions resulting from the discretization of Eqs. (3) and (4) (Bartosh et al., 2024) .Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: In Section 6, we discuss the limitations of the proposed approach, and of the experimental validation. Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
equivariant neural diffusion: 
 We now introduce Equivariant Neural Diffusion (END), that generalizes the Equivariant Diffusion Model (EDM) (Hoogeboom et al., 2022) , by defining the forward process through a learnable transformation. Our approach is a synthesis of NFDM introduced in Section 2.3, and leverages ideas of EDM outlined in Section 2.2 to maintain the desired invariance of the learned marginal distribution p θ,φ (z 0 ). By providing an equivariant learnable transformation F φ and an equivariant data point predictor xθ , we show that it is possible to obtain a generative model with the desired properties. Finally, we propose a simple yet flexible parameterization meeting these requirements.The key innovation in END lies in its forward process, which is also leveraged in the reverse (generative) process. The forward process is defined through a learnable time-and data-dependent transformation F φ (ε, t, x), such that the latent z t transforms covariantly with the injected noise ε (i.e. a collection of random vectors) and the data point x,F φ (Rε, t, Rx) = RF φ (ε, t, x) = Rz t , ∀ R ∈ O(3).We then define xθ as another learnable equivariant function, such that the predicted data point transforms covariantly with the latent variable z t , i.e. xθ (Rz t , t) = R xθ (z t , t). Finally, we choose the noise and prior distribution, i.e. p(ε) and p(z 1 ), to be invariant to the considered symmetry group.Invariance of the learned distribution With the following choices: (1) p(z 1 ) an invariant distribution, (2) F φ an equivariant function that satisfies F φ (Rε, t, Rx) = RF φ (ε, t, x), and (3) xθ an equivariant function, we have that the learned marginal p θ,φ (z 0 ) is invariant as desired. This can be shown by demonstrating that the reverse SDE is equivariant. We start by noting that the reverse SDE in END is given bydz t = fθ,φ (z t , t) dt +g φ (t) d w .As the Wiener process is isotropic, this boils down to verifying that the drift term, fθ,φ (z t , t) is equivariant, i.e. fθ,φ (Rz t , t) = R fθ,φ (z t , t). As the drift is expressed as a sum of two terms, we inspect each of them separately. The first term isf φ z t , t, xθ (z t , t) = ∂F φ ε, t, xθ (z t , t) ∂t ε=F -1 φ (zt,t, xθ (zt,t)).If F φ is equivariant, then so is its time-derivative (see Appendix A.3.1). The same holds for its inverse with respect to ε (see Appendix A.3.2), such that we have F -1 φ (Rz t , t, Rx) = RF -1 φ (z t , t, x) = Rε. We additionally have that xθ is equivariant, by definition. As the equivariance of F φ implies the equivariance of q φ , from the second term of the drift, we observe that, for y t = Rz t , we have∇ yt log q φ y t | xθ (y t , t) = R∇ zt log q φ z t | xθ (z t , t) , ∀ R ∈ O(3).In summary, in addition to an invariant prior, an equivariant F φ and an equivariant xθ ensure the equivariance of the reverse process, and hence the invariance of the learned distribution. In Appendix A.3.3, we additionally show that the objective function is invariant, i.e. L END (Rx) = L END (x), ∀R ∈ O(3).We note that alternative formulations for the drift of the reverse process, fθ,φ , exist. Most notably, it can be learned directly through an equivariant function, without explicit dependence on F φ , while maintaining the desired invariance.We now introduce a simple parameterization of F φ that meets the requirements outlined above,EQUATIONwhere, due to the geometric nature of x, U φ (x, t) ∈ R (M -1)×3×3 is structured as a block-diagonal matrix where each block is a 3 × 3 matrix, ensuring a cheap calculation of the inverse transformation and its Jacobian. This is equivalent to a diagonal parameterization in the case of scalar features.Similarly to EDM, our parametrization of F φ leads to a conditional marginal q φ (z t |x) that is a conditional Gaussian with (block-) diagonal covariance, with the notable difference that the mean and covariance are now data-and time-dependent, and learnable through F φ ,EQUATIONwhereΣ φ (x, t) = U φ (x, t)U ⊤ φ (x, t), such that Σ φ (x, t) is also block-diagonal.As F φ is linear in ε, both µ φ and U φ must be equivariant functions whose outputs transform covariantly with x, in order to ensure the desired equivariance of F φ ,F φ (Rε, t, Rx) = µ φ (Rx, t) + U φ (Rx, t)Rε = Rµ φ (x, t) + R U φ (x, t)ε = RF φ (ε, t, x).We can then readily check that the resulting q φ is equivariant, as ∀ R ∈ O(3), we have thatq φ (z t |x) = N z t |µ φ (x, t), Σ φ (x, t) = N Rz t |Rµ φ (x, t), RΣ φ (x, t)R ⊤ = q φ (Rz t |Rx).We note that other, and more advanced, parametrizations are possible, e.g. based on normalizing flows with a flow architecture similar to that of Klein et al. (2024) .Prior and Reconstruction While not strictly required, it can be advantageous to parameterize the transformation F φ such that the prior and reconstruction losses need not be computed. To do so, we design F φ (ε, x, t) such that the conditional distribution evolves from a low-variance Gaussian centered around the data, i.e. q(z 0 |x) ≈ N (z 0 |x, δ 2 I) to an uninformative prior distribution (that contains no information about the data distribution), i.e. a unit Gaussian q(z 1 |x) ≈ N (z 1 |0, I). Specifically, we parameterize F φ through the following functions,EQUATIONEQUATIONwhich ensure that (i) in t = 0, µ φ (x, 0) = x and Σ φ (x, 0) = δ 2 I; (ii) in t = 1, µ φ (x, 1) = 0 and Σ φ (x, 1) = I; while being unconstrained for t ∈]0, 1[. We note that this is only one possible parametrization for F φ , and that, by adapting µ φ and U φ , richer priors can easily be leveraged -e.g. harmonic prior given a molecular graph or scale-dependent prior (Jing et al., 2023; Irwin et al., 2024) .Implementation In practice, F φ is implemented as a neural network with an architecture similar to that of the data point predictor xθ (z t , t), but with a specific readout layer that produces[μ φ (x, t), σφ (x, t), Ūφ (x, t)]. The mean output μφ (x, t) is similar to that of xθ (z t , t). For U φ (x, t), as Σ φ (x, t) = U φ (x, t)U ⊤ φ (x, t) should rotate properly, σφ (x, t) is a positive invariant scalar, while Ūφ (x, t) is constructed as a matrix whose columns are vectors that transform covariantly with x.To ease notation, we introduced all notations in the linear subspace R, however in practice we work in the ambient space, i.e. r ∈ R M ×3 . We detail in Appendix A.5.1, how working in ambient space is possible. The training and sampling procedures are detailed in Algorithms 1 and 2 in the appendix.While unconditional generation is a required stepping stone, many practical applications require some form of controllability. As other generative models, DMs can model conditional distributions p(x|c), where c is a given condition. Different methods exist for sampling from the conditional distribution, e.g. Wu et al. (2024) , but the simplest approach consists in training a conditional model on pairs (x, c). In such setting, F φ and xθ simply receive an extra input c representing the conditional information, such that they respectively become F φ (ε, t, x, c), and xθ (z t , t, c). It is important to note that, compared to conventional DMs, the forward process of END is now also condition-dependent.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
experiments: 
 In this section, we demonstrate the benefits of END with a comprehensive set of experiments. In Section 4.1, we first display the advantages of END for unconditional generation on 2 standard benchmarks, namely QM9 (Ramakrishnan et al., 2014) and GEOM-DRUGS (Axelrod and Gomez-Bombarelli, 2022) . Then, in Section 4.2, we perform conditional generation in 2 distinct settings on QM9. Additional experimental details are provided in Appendix A.6.Datasets The QM9 dataset (Ramakrishnan et al., 2014) contains 134 thousand small-and mediumsized organic molecules with up to 9 heavy atoms, and up to 29 when counting hydrogen atoms. GEOM-DRUGS (Axelrod and Gomez-Bombarelli, 2022) contains 430 thousand medium-and largesized drug-like molecules with 44 atoms on average, and up to maximum 181 atoms. We use the same data setup as in previous work (Hoogeboom et al., 2022; Xu et al., 2022) .Task We sample 10 000 molecules using the stochastic sampling procedure detailed in Algorithm 2. As END is trained in continuous-time, we vary the number of integration steps from 50 to 1000. We repeat each sampling for 3 seeds, and report averages along with standard deviations for each metric.We follow previous work (Hoogeboom et al., 2022; Xu et al., 2023) , and first evaluate the chemical quality of the generated samples in terms of stability, validity, and uniqueness (in Tables 1, 6 and 9 ). On QM9, we additionally evaluate how well the model learns the atom and bond types distributions by measuring the total variation between the dataset's and generated distributions, as well as the overall quality of the generated structures via their strain energy, expressed as the energy difference between the generated structure and a relaxed version thereof (in Tables 2 and 6 ). On GEOM-DRUGS, we additionally compute connectivity, total variation for atom types, and strain energy (in Tables 2 and 9 ). Connectivity accounts for the fact that validity can easily be increased by generating several disconnected fragments (where only the largest counts towards Table 1 : Stability and validity results on QM9 and GEOM-DRUGS obtained over 10000 samples, with mean/standard deviation across 3 sampling runs. END compares favorably to the baseline across all metrics on both datasets, while offering competitive performance for reduced number of sampling steps. It reaches a performance level similar to that of current SOTA methods. ‡ denotes results obtained by our own experiments.GEOM-DRUGS validity), the total variation ensures that the model properly samples all atom types, whereas the strain energy evaluates the generated geometries. On both datasets, we also report additional drug-related metrics as per Qiang et al. (2023) (in Tables 7 and 10 ). More details about evaluation metrics are provided in Appendix A.6.1.Stability (↑) Val. / Uniq. (↑) Stability (↑) Val. / Conn. (↑) A [%] M [%] V [%] V×U [%] A [%] V [%] V×C [%] ModelBaselines We compare END to several relevant baselines from the literature: the original EDM (Hoogeboom et al., 2022) ; EDM-BRIDGE (Wu et al., 2022) , an improved version of EDM that adds a physics-inspired force guidance in the reverse process; GEOLDM (Xu et al., 2023) , an equivariant latent DM; and GEOBFN (Song et al., 2024) , a geometric version of Bayesian Flow Networks (Graves et al., 2024) . A detailed discussion about related work can be found in Section 5.Ablations of END In addition to baselines from the literature, we compare different ablated versions of END. As the key component to our method is the learnable forward process, the logical ablation is whether to include a learnable forward (=END), or not (=EDM). To ensure a fair comparison and that any difference in performance does not stem from an increase in learnable parameters, an architectural change or the training paradigm, we implement our own EDM (Hoogeboom et al., 2022) , denoted EDM*. It features the exact same architecture as END, the same amount of learnable parameters (i.e. through a deeper xθ ), and is similarly trained in continuous-time. Additionally, we provide EDM* + γ, similar to EDM* but with a learned SNR (Kingma et al., 2021) for each data modality (i.e. atomic types and coordinates), and END (µ φ only), an ablated version of END where only the mean is learned whereas the standard deviation of the conditional marginal is pre-specified and derived from the noise schedule of EDM. Table 5 provides an overview of the compared models.Results on QM9 Our main results on the QM9 dataset are summarized in the left part of Tables 1 and 2 , where END is shown to significantly outperform the baseline and reach a level of performance similar to current SOTA methods, GEOLDM (Xu et al., 2023) and GEOBFN (Song et al., 2024) , across stability and validity metrics. The ablation study in Appendix A.2.1 clearly reveals the benefits of a learnable forward. On the one hand, the two variants of END are shown to outperform (or be on par with) all baselines across all metrics in Table 6 (in particular, in terms of the more challenging molecule stability), and be in better agreement with the data distribution in Table 7 (except for QED which is captured perfectly by all methods). On the other hand, we observe that, with as few as 100 integration steps, END is capable of generating samples that are qualitatively better than those generated by the simpler baselines in 1000 steps. A few illustrative QM9-like molecules generated by END are displayed in Fig. 1 .Results on GEOM-DRUGS Our main results on the more realistic GEOM-DRUGS dataset are presented in the right part of Tables 1 and 2 , where we observe that END is competitive against the compared baselines in terms of atom stability, while being slightly subpar in terms of validity. However, when accounting for connectivity (via the V×C metric), we observe that END does outperform the baseline with an increased sucess rate of around 20% on average, as well as better than the SOTA method GEOLDM (Xu et al., 2023) by a large margin. The generated molecules also better follow the atom types distribution of the dataset -as per the lower total variation, and feature better geometries than those generated by competitors -as implied by the lower strain energy. As for QM9, the ablation study provided in Appendix A.2.3 illustrates the clear improvement provided by a learnable forward process compared to a fixed one. In this more challenging setting, only learning the mean function yields a slight decrease in performance across the metrics reported in Table 9 , and a similar agreement with the dataset in terms of the drug-related metrics in Table 10 , compared to the full model. Furthermore, while each sampling step is currently slower than EDM* (Table 11 ), the improved sampling efficiency afforded by END (in terms of integration steps) allows practical time gains on this more complex dataset. As a concrete example, samples obtained with END with only 100 steps are qualitatively better those generated by EDM* in 1000 integration steps, amounting to a 3x time cut. Examples of GEOM-DRUGS-like molecules generated by END are provided in Fig. 1 . 3 , where we observe that CEND significantly outperforms the baseline, and offers nearly fully controllable composition generation. Additionally, reducing the number of sampling steps has a very limited impact on the controllability. Finally, we perform an ablation whose results are presented in Table 8 , where fixing the standard deviation is shown to lead to a small decrease in performance, with respect to the full model while remaining significantly better than the baseline with fixed forward.Task 2: substructure-conditioned generation We adopt a setup similar to that of Bao et al. (2023) , and train a conditional END, where the condition is a molecular fingerprint encoding structural information about the molecule. A fingerprint is a binary vector c = (c 1 , ..., c F ) ∈ {0, 1} F , where c f is set to 1 if substructure f is present in the molecule, or to 0 if not. Fingerprints are obtained using OPENBA-BEL (O'Boyle et al., 2011) . We evaluate the ability of the compared models to leverage the provided structural information, by sampling conditionally on fingerprints obtained from the test set. We then compute the Tanimoto similarity between the fingerprints yielded by generated molecules and the fingerprints provided as conditional inputs. We compare CEND to EEGSDE (Bao et al., 2023) , an improved version of EDM (Hoogeboom et al., 2022) , that performs conditional generation by combining a conditional diffusion model and regressor guidance. Our results are presented in Table 4 , along with a handful of samples in Fig. 2 , where CEND is shown to offer better controllability than the compared baselines, as highlighted by the higher similarity. 
related work: 
 The main approaches to molecule generation in 3D are autoregressive models (Gebauer et al., 2019; Simm et al., 2020; Gebauer et al., 2022; Luo and Ji, 2022; Daigavane et al., 2024) , flow-based models (Garcia Satorras et al., 2021) , and diffusion models (Hoogeboom et al., 2022; Igashov et al., 2024) . A notable exception to the geometric graph representation of 3D molecules are voxels (Skalic et al., 2019; Ragoza et al., 2022; O Pinheiro et al., 2024) , from which the 3D graph is extracted via some post-processing procedure. Recently, several works have shown that leveraging 2D connectivity information can lead to improved results (Peng et al., 2023; Vignac et al., 2023; Le et al., 2024) . While not incompatible with END, we perform our experiments without modeling that auxiliary information, and therefore do not compare to these approaches directly. Other frameworks have also been tailored to molecule generation, such as Flow Matching (Lipman et al., 2022; Song et al., 2023; Irwin et al., 2024) or Bayesian Flow Networks (Graves et al., 2024; Song et al., 2024) , also showing promises for accelerated sampling.In the realm of diffusion models for molecules, EDM-BRIDGE (Wu et al., 2022) and EEGSDE (Bao et al., 2023) extend upon a continuous-time formulation of EDM (Hoogeboom et al., 2022) , as END also does. Based on the observation that there exists an infinity of processes mapping from prior to target distributions, EDM-BRIDGE constructs one such process that incorporates some prior knowledge, i.e. part of the drift term is a physically-inspired force term. END can be seen as a generalization of EDM-BRIDGE, where the forward drift term is now learned instead of prespecified. Through experiments, we show that a learnable forward performs better than a fixed one, even when the latter is physics-inspired. EEGSDE specifically targets conditional generation by combining (1) a conditional model score model, (2) a method similar to classifier-guidance (requiring the training of an auxiliary model). In CEND, we instead only learn a conditional model. Finally, GEOLDM (Xu et al., 2023) is a latent diffusion model that performs diffusion in the latent space of an equivariant Variational Auto-Encoder (VAE), and it can be seen as a particular case of END, whereF φ (ε, t, x) = α t E φ (x) + ϵσ t , with E φ (x)denoting the (time-independent) encoder of the VAE. 
conclusion: 
 In this work, we have presented Equivariant Neural Diffusion (END), a novel diffusion model that is equivariant to Euclidean transformations. The key innovation in END lies in the forward process that is specified by a learnable data-and time-dependent transformation. Experimental results demonstrate the benefits of our method. In the unconditional setting, we show that END yields competitive generative performance across two different benchmarks. In the conditional setting, END offers improved controllability, when conditioning on composition and substructure. Finally, as a by-product of the introduced learnable forward, we also find the sampling efficiency (in terms of integration steps) to be improved, while that property is not actively optimized for in the design of the parameterization nor in the training procedure.Avenues for future work are numerous. In particular, further leveraging the flexible framework of NFDM (Bartosh et al., 2024) to constrain the generative trajectories, e.g. to be straight and enable even faster sampling, modelling bond information, or extending the conditional setting to other types of conditioning information, e.g. other point cloud or target property, are all promising directions.Limitations From a computational perspective, END is currently slower to train and sample from compared to a baseline with fixed forward with identical number of learnable parameters. Even if convergence is similarly fast, each training (resp. sampling) step incurs a relative ≈ 2.5x (resp. ≈ 3x). However, END usually requires much fewer number of function evaluations to achieve comparable (or better) accuracy, and alternative (and more efficient) parameterizations of the reverse process exist. In particular, the drift fθ,φ could be learned without direct dependence on f φ , thereby leading to an improved training time and, more importantly, a very limited overhead with respect to vanilla DMs for sampling. In terms of scalability, END suffers from limitations similar to concurrent approaches. It operates on fully-connected graphs -preventing its scaling to very large graphs, and models categorical features through an arbitrary continuous relaxation -potentially suboptimal and scaling linearly with the number of chemical elements in the modeled dataset. Encodings that scale more gracefully, such as that of Analog Bits (Chen et al., 2023) (logarithmic) or GeoLDM (Xu et al., 2023) (learned low-dimensional), are good candidates to better deal with discrete features within END. In terms of data, the presented findings are limited to organic molecules and the metrics, while widely used in the community, also have some evident limitations. To fully assess the practical interest of the generated molecules, thorough validation with QM simulations would be required.Broader Impact Generative models for molecules have the potential to accelerate in-silico discovery and design of drugs or materials. This work proposes an instance of such model. As any generative model, it also comes with potential dangers, as this could be misused for designing e.g. chemicals with adversarial properties.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? Answer: [Yes] Justification: The data configuration follows from previous work, while most of the experimental details are provided in Appendix.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] Justification: All sampling results are averages over multiple seeds and we include standard errors.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA] Justification: The paper does not pose such risks.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: We have cited the original papers that released the datasets.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. 
 
 Appendix: A.1 Compared modelsIn Table 5 , we detail all the compared diffusion models in terms of their transformation F φ . f : X × [0, 1] → Y be a function that is equivariant to actions of the group O(3), such that R • f (x, t) = f (R • x, t), ∀R ∈ O(3). Proof sketch We need to show that ∂ ∂t f (R • x, t) = R • ∂ ∂t f (x, t) , ∀R ∈ O(3) and ∀x ∈ X . ∂ ∂t f (R • x, t) = ∂ ∂t R • f (x, t) , = R • ∂ ∂t f (x, t) ,where the last equality follows by linearity.Let f : X → Y be a function that (1) is equivariant to the action of the group O(3), and (2) admits an inverse f -1 : Y → X , then f -1 is also equivariant to the action of O(3).Proof sketch We need to show that f-1 R • y = R • f -1 (y), ∀R ∈ O(3) and ∀y ∈ Y.Since f is invertible, we have that to any y ∈ Y corresponds a unique x ∈ X , such that y = f (x), and thatf -1 (y) = f -1 (f (x)) = x. As f is equivariant to the action of O(3), we have that R • f (x) = f (R • x), ∀R ∈ O(3): f -1 R • y = f -1 R • f (x) , = f -1 f (R • x) , = R • x, = R • f -1 (y).In this section, we show that the objective function is invariant under the action of O(3): 3), provided that F φ and xθ are equivariant.L END (Rx) = L END (x), ∀R ∈ O(L END (Rx) = E u(t),qφ(zt|Rx)q(Rx) 1 2g 2 φ (t) f φ (z t , t, Rx) - g 2 φ (t) 2 ∇ zt log q φ (z t |Rx) -f φ (z t , t, xθ (z t , t)) + g 2 φ (t) 2 ∇ zt log q φ (z t | xθ (z t , t)) 2 2 , = E u(t),qφ(zt|Rx)q(Rx) 1 2g 2 φ (t) f φ (RR -1 z t , t, Rx) - g 2 φ (t) 2 ∇ zt log q φ (RR -1 z t |Rx) -f φ (RR -1 z t , t, xθ (RR -1 z t , t)) + g 2 φ (t) 2 ∇ zt log q φ (RR -1 z t | xθ (RR -1 z t , t)) 2 2 , = E u(t),qφ(zt|Rx)q(Rx) 1 2g 2 φ (t) Rf φ (R -1 z t , t, x) - g 2 φ (t) 2 ∇ zt log q φ (R -1 z t |x) -Rf φ (R -1 z t , t, xθ (R -1 z t , t)) + g 2 φ (t) 2 ∇ zt log q φ (R -1 z t | xθ (R -1 z t , t)) 2 2 , = E u(t),qφ(Ryt|Rx)q(Rx) 1 2g 2 φ (t) Rf φ (y t , t, x) - g 2 φ (t) 2 R∇ yt log q φ (y t |x) -Rf φ (y t , t, xθ (y t , t)) + g 2 φ (t) 2 R∇ yt log q φ (y t | xθ (y t , t)) 2 2 , = E u(t),qφ(yt|x)q(x) 1 2g 2 φ (t) f φ (y t , t, x) - g 2 φ (t) 2 ∇ yt log q φ (y t |x) -f φ (y t , t, xθ (y t , t)) + g 2 φ (t) 2 ∇ yt log q φ (y t | xθ (y t , t)) 2 2 , = L END (x).The first equality is obtained by replacing x by Rx in the definition of the objective function Eq. ( 5). The second is obtained by multiplying by RR -1 = I. The third equality by leveraging that f φ , q φ and xθ are equivariant. We then perform a change of variable y t = R -1 z t . As rotation does preserve distances, we obtain the last equality.Require: q(x), F φ , xθ for training iterations do x ∼ q(x), t ∼ u(t), ε ∼ p(ε) z t ← µ φ (x, t) + U φ (x, t)ε L = 1 2g 2 φ (t) f B φ (z t , t, x) -fθ,φ (z t , t) 2 2Gradient step on θ and φ end for Algorithm 2 Stochastic sampling from END Require: F φ , xθ , integration steps T , empirical distribution of number of atoms p(N) ∆t = 1 T N ∼ p(N ) z 1 ∼ p(z 1 ) for t = 1, ..., 1 T do w ∼ N (0, I) z t-∆t ← z t -fθ,φ (z t , t)∆t + g φ (t) w√ ∆t end for x ∼ p(x|z 0 ) A.5 Details about F φ A.5.For simplicity, we omitted in Section 3 and Appendix A.5.1 that molecules are described as tuples: x = (r, h), as only r transform under Euclidean transformations. For the invariant features h, we use the following parameterizationEQUATIONEQUATIONwhich ensures that µ Implementation As described in the main text in Section 3.2, F φ is implemented as a neural network with an architecture similar to that of the data point predictor xθ (z t , t), but with a specific readout layer that produces the outputs related to r ([μ φ (x, t), σφ , Ūφ (x, t)]). Additionally, it produces μ(h) φ (x, t) and σ(h) φ (x, t) as invariant outputs.Inverse transformation The logarithm of the determinant of the inverse transformation logEQUATIONwhere V is defined as in Eq. ( 18).In addition to the details provided in this section, we release a public code repository with our implementation of END.In this section, we describe the metrics employed to evaluate the different models:• Stability: An atom is deemed stable if it has a charge of 0, whereas a molecule is stable if all its atoms have 0 charge. We reuse the lookup table from Hoogeboom et al. (2022) to infer bond types from pairwise distances.• Validity and Connectivity: Validity corresponds to the percentage of samples that can be parsed and sanitized by rdkit (Landrum et al., 2013) , after inference of the bonds using the lookup table mechanism (Hoogeboom et al., 2022) . It should be noted that the metric does not penalize fragmented samples as long as each individual fragment appears valid. This can be problematic when running evaluation on larger compounds such as those in GEOM-DRUGS, as models tend to generated disconnected structures. To account for that, we also report Connectivity, which simply check that a valid molecule is composed of exactly one fragment.• Uniqueness: Uniqueness is expressed as the proportion of samples that are valid and have a unique SMILES string (Weininger, 1988) among all the generated samples. On GEOM-DRUGS, we do not report Uniqueness as all generated samples appear unique (as per previous work).• Total variation: The total variation is computed as the MAE between the (discrete) marginal obtained on the training data and on the generated samples. For bond types on QM9, we compute the ground truth and generated distributions using the lookup table mechanism (Hoogeboom et al., 2022) .• Strain Energy: The strain energy is expressed as the difference in energy between generated structures and a relaxed version thereof obtained as per rdkit's MMFF (Landrum et al., 2013) . From the generated samples, we infer rdkit mol objects using OPENBABEL (O'Boyle et al., 2011) . We only evaluate the strain energy of valid and connected samples.• SA, QED, log P and MW: SA denotes the "Synthetic Accessibility Score", which is a rule-based scoring function that evaluates the complexity of synthesizing a structure by organic reactions (Ertl and Schuffenhauer, 2009) . We normalized its values between 0 and 1, with 0 being "difficult to synthesize" and 1 "easy to synthesize". QED denotes "Quantitative Estimation of Drug-likeness". log P denotes the octanol-water partition coefficient. MW denotes the molecular weight. We employ the rdkit's implementation of all metrics. To do so, we convert the generated samples to rdkit mol objects using OPENBABEL (O'Boyle et al., 2011) . We then only evaluate the different metrics for valid and connected samples.• MMD: On QM9, we follow the procedure of Daigavane et al. (2024) , and compute the MMD Gretton et al. (2012) between true and generated pairwise distances distributions for the 10 most common bonds in the dataset: ["C-H:1.0", "C-C:1.0", "C-O:1.0", "C-N:1.0", "H-N:1.0", "C-O:2.0", "C-N:1.5", "H-O:1.0", "C-C:1.5", "C-N:2.0"].Our forward transformation F φ and data point predictor xθ share a common neural network architecture that we detail here. The architecture is similar to that of EQCAT (Le et al., 2022) , and updates a collection of invariant and equivariant features for each node in the graph. We choose that architecture because it allows for an easy construction of Ūφ (x, t) by linear projection of the final equivariant layer.We follow previous work (Hoogeboom et al., 2022) and consider fully-connected graphs. We initially featurize pairwise distances through Gaussian Radial Basis functions, with dataset-specific cutoff taken large enough to ensure full connectivity. In opposition to Hoogeboom et al. (2022) , we do not update positions in the message-passing phase, but instead obtain the positions prediction through a linear projection of the final equivariant hidden states. The predictions for the invariant features are obtained by reading out the final invariant hidden states.Optimization For all model variants, we employ Adam with a learning rate of 10 -4 . We perform gradient clipping (norm) with a value of 10 on QM9, and a value of 1 on GEOM-DRUGS.We reuse the data setup from previous work (Hoogeboom et al., 2022; Xu et al., 2023) .QM9 On QM9, we use 10 layers of message passing for EDM*, while the variants of END feature 5 layers of message-passing in F φ and 5 layers in xθ . For all models, we use 256 invariant and 256 equivariant hidden features, along with an RBF expansion of dimension 64 with a cutoff of 10Å for pairwise distances. This ensures that the compared models have the same number of learnable parameters, i.e. ≈ 9.4M each. We train all models for at most 1000 epochs with a batch size of 64.GEOM-Drugs On GEOM-DRUGS, we use 10 layers of message passing for EDM*, while the variants of END feature 5 layers of message-passing in F φ and 5 in xθ . The hidden size of the invariant and equivariant features is set to 192, along with an RBF expansion of dimension 64 with a cutoff of 30Å for pairwise distances (as to ensure full-connectivity). Each model features ≈ 5.4M learnable parameters. We train all models for 10 epochs with an effective batch size of 64.We use 10 layers of message passing for EDM*, while the variants of END feature 5 layers of message-passing in F φ and 5 in xθ . The hidden size of the invariant and equivariant features is set to 192 , along with an RBF expansion of dimension 64 with a cutoff of 10Å for pairwise distances. We train all models for 1000 epochs with a batch size of 64.After an initial encoding, the conditional information is introduced at the end of each message passing step, and alters the scalar hidden states through a one-layer MLP, that shares the same dimension as the hidden scalar state.Composition-conditioned generation The encoding of the condition follows that of Gebauer et al. (2022) . Each atom type gets its own embedding (of dimension 64), weighted by the proportion it represents in the provided formula. The weighted embeddings of all atom types are then concatenated and flattened, and the obtained vector (of dimension 64×number of atom types ) is processed through a 2-layer MLP with 64 hidden units.The compositions used at sampling time are extracted from the validation and test sets. For each unique formula, the model gets to generate 10 samples. The reported matching rate refers to the % of generated samples featuring the prompted composition.The encoding of the condition follows that of Bao et al. (2023) The model is evaluated by computing the Tanimoto similarity between the fingerprints obtained from the generated samples and the fingerprints provided as conditional inputs.All experiments were run on a single GPU. The experiments on QM9 were run on a NVIDIA SM3090 with 24 GB of memory. The experiments on GEOM-DRUGS were run on NVIDIA A100 with 40 GB. Training took up to 7 days.Sampling The current implementation of END leads to ≈ 3x increase relative to EDM (with comparable number of learnable parameters) per function evaluation when performing sampling.However, END usually requires much fewer number of function evaluations to achieve comparable (or better) accuracy, and we note that alternative parameterizations of the reverse process are possible.In particular, the drift of the reverse process fθ,φ could be learned without direct dependence on f φ , thereby leading to very limited overhead with respect to vanilla diffusion models for sampling -only one neural network forward network would then be required. For reference, we report sampling times on QM9 (1024 samples) in Table 11 , for varying numbers of integration steps. Training A training step on QM9 (common batch size of 64) takes on average ≈ 0.37s (END) vs 0.16s (EDM), this corresponds to a ≈ 2.3x relative increase. We observe the same trend on GEOM-Drugs (with an effective batch size of 64), a training step takes on average ≈ 0.40s for END vs. ≈ 0.15s for EDM (corresponding to a ≈ 2.7x relative increase). In summary, other things equal, END leads to ≈ 2.5x increase relative to EDM per training step while we find it to converge with a similar number of epochs. As for sampling, a direct parametrization of fθ,φ would enable faster training -while still requiring the evaluation of f φ to obtain the target reverse drift term in Eq. ( 5).1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? Answer: [Yes] Justification: The method section shows that the proposed model has the properties we claim it has, while the experimental section demonstrates its performance. We additionally highlight limitations Section 6. Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 


Decision: The reviewers all agreed that the method performs well (esp. in challenging conditions), and most reviewers noted that the paper is well written. The method is a novel combination of existing ideas, and different reviewers appraised this differently, with some finding it somewhat incremental (e.g. hMME) while others (e.g. K86T) finding it a creative combination. Many minor concerns raised by the reviewers were effectively addressed during the discussion, and in the end only hMME remained skeptical. Their primary concern is a comparison to the recent work GeoBFN. The authors have now shown their method to be competitive, and are working on running comparisons with additional metrics as requested.

In my view the method is sufficiently novel, and already fairly well benchmarked in the initial submission. The authors have been very active improving the paper and addressing reviewer concerns during the discussion period, and overall this now looks like a solid contribution. So I recommend the paper for acceptance at NeurIPS.

Reviews: review: paper_summary: paper_summary: This paper proposes an extension of diffusion models dubbed Equivariant Neural Diffusion, which leverages  a learnable forward diffusion process to enhance flexibility. The entire framework has been constructed such that physical symmetry, i.e., equivariance/invariance, of the density is preserved. Experiments are performed on QM9 and DRUGS datasets in the task of molecule generation from scratch as well as controllable generation.
strength_weakness: strength_weakness: 1. The proposed approach seems to be an incremental combination of geometric diffusion models and neural flow diffusion models. The way of combining these two flavors incurs limited technical novelty. The core design mostly lies in constructing $F\_\varphi$ which is equivariant.

2. The performance on QM9 and DRUGS is a bit marginal compared with the selected baselines.

3. Missing important baselines, e.g., GeoBFN [1], which has shown strong performance on the same task, i.e., molecule generation. Moreover, GeoBFN can also achieve high performance with very few sampling steps, even only 20.

[1] Song et al. Unified generative modeling of 3d molecules with bayesian flow networks. In ICLR'24.
questions: questions: Q1. How does the method perform compared with GeoBFN, especially with different number of sampling steps?

Q2. Could the method be applied to systems with larger scales, e.g., proteins?
limitations: limitations: The authors have discussed the limitations, including scaling, limited application scope, etc.
paper_summary: paper_summary: The Equivariant Neural Diffusion (END) model is a novel approach for molecule generation in 3D that maintains equivariance to Euclidean transformations. Unlike traditional diffusion models that use a pre-specified forward process, END introduces a learnable forward process, parameterized through a time- and data-dependent transformation. This innovation allows the model to adapt better to the underlying data distribution. Experimental results demonstrate that END outperforms strong baselines on standard benchmarks for both unconditional and conditional generation tasks, particularly excelling in generating molecules with specific compositions and substructures. This flexibility in modeling complex molecular structures suggests significant potential for applications in drug discovery and materials design.
strength_weakness: strength_weakness: **Performance Consistency:** The performance of END is not consistently superior to existing baselines. Although it shows competitive results, there are instances where traditional models, like EDM and its variants, outperform END, particularly in metrics such as validity and uniqueness across different datasets.

**Complexity and Scalability:** The added complexity of a learnable forward process, while innovative, increases the model’s training time and resource requirements. END requires more computational resources and longer training periods compared to simpler, fixed-process models like EDM . The model operates on fully-connected graphs, which limits its scalability to larger datasets and more complex molecular structures. This constraint can hinder its applicability in more demanding real-world scenarios.
questions: questions: - Why was GeoDiff not included as a baseline in your comparisons? 
- Could you provide more detailed ablation studies that isolate the contributions of the learnable forward process and other key components of END?
limitations: limitations: **Computational Complexity:** The learnable forward process in the END model increases computational complexity, resulting in longer training times and higher resource usage. The authors should discuss potential optimizations to reduce this overhead, such as more efficient algorithms or hybrid approaches. Comparing training times and resource requirements with simpler models would provide useful insights into the model's efficiency.

**Scalability Issues:** END's architecture limits its scalability to larger datasets and more complex molecular structures like proteins. To enhance scalability, the authors should explore strategies like sparse representations or hierarchical approaches. These methods could enable the model to handle larger and more complex datasets effectively.
paper_summary: paper_summary: Paper presents, END, a diffusion models for 3D molecule generation that
    
- is equivariant to euclidean transformations and 

- includes a learnable forward process.
  
 Specifically, the forward process in the presented models, is defined as a learnable transformation, dependent on both time and data such that the resulting latent representation $z_t$ transforms covariantly with the injected noise. This is the main difference between the forward pass of Neural Flow Diffusion Models and the proposed method.
strength_weakness: strength_weakness: - Lack of a thorough ablation of the proposed method.
questions: questions: - The only form of ablation i see for the proposed model is in Table 1 where two versions of END are provided. However even here, it seems the END with $\mu_\phi$ performs similar or sometimes better (according to the presented metrics) than the full END model. Is the same pattern observed for the conditional generation tasks?
 
- Relating to the first question above, can a much more thorough ablation of the proposed method be provided in both testing scenarios to really ascertain the utility of the components in the proposed method?

- Can the training times for the benchmarked methods be provided for comparison? While it is stated passingly that END requires more training time, can this be cast in contrast with the baselines by providing the actual numbers?
limitations: limitations: Limitations are adequately discussed by the authors.
paper_summary: paper_summary: This paper presents Equivariant Neural Diffusion (END), a novel diffusion model for 3D molecule generation. The major novelty of END over previous molecule diffusion models lies in adopting a learnable forward process based on neural flow diffusion models. Experiments show that END can achieve good performance on benchmark datasets.
strength_weakness: strength_weakness: - The novelty contribution of the proposed END method is not high, as END is largely a combination of EDM [1] and neural flow diffusion models. Particularly, the paper does not give a clear discussion or analysis about why adopting learnable forward diffusion process is useful and beneficial to 3D molecule generation, or what molecular structures can be additionally captured by END through learnable forward diffusion process compared with previous diffusion models.
- There already exist some SDE based 3D molecule generation methods like EDM-BRIDGE [2] and EEGSDE [3]. Authors are encouraged to highlight the key difference in the diffusion process between END and these methods.
- Compared with GEOLDM, END does not show better performance (Table 1 and 2), which weakens the claim about the advantages of using learnable forward process. Since the main evaluation metrics proposed by EDM [1] in 3D molecule generation are saturating in recent literatures, authors are encouraged to adopt metrics proposed by HierDiff [4] to better evaluate the quality of generated 3D molecules.

[1] Equivariant Diffusion for Molecule Generation in 3D. ICML 2022.  
[2] Diffusion-based Molecule Generation with Informative Prior Bridges. NeurIPS 2022.  
[3] Equivariant Energy-Guided SDE for Inverse Molecular Design. ICLR 2023.  
[4] Coarse-to-Fine: a Hierarchical Diffusion Model for Molecule Generation in 3D. ICML 2023.

-------------------Post Rebuttal---------------------

I appreciate authors' efforts in addressing my concerns and questions in rebuttal. After reading over authors' rebuttal responses and pdf, I think all my concerns have been addressed so I increased my score. I hope authors will carefully add all rebuttal updates (discussion, analysis and experiment results) to the revised version of paper in the future.
questions: questions: See Weaknesses part.
limitations: limitations: Yes.

Comments: title: Follow up ablation on GEOM-Drugs
Follow up
Follow up
Follow-up
Follow-up Response
comment: We want to thank again the reviewer for suggesting us to run additional ablations. 

As promised in our initial rebuttal, we provide here additional results on the more challenging GEOM-Drugs -- i.e. an ablated model where only the mean is learned (the standard deviation of the conditional marginal is pre-specified and derived from the same noise schedule as EDM). 

The results are collected in the last row of the table herebelow, and presented with the initial results for better readability. 
Similarly to the conditional setting, learning only the mean provides a clear improvement compared to the baseline, but is shown to perform slightly worse to the full model across all metrics except validity.



|                 |       | At. Stab. [\%]                      | V [\%]                      | V$\times$C [\%]             | TV$_A$ [$10^{-2}$]               |
|-----------------|-------|-----------------------------|-----------------------------|-----------------------------|-----------------------------|
| Model           | Steps |                             |                             |                             |                             |
| EDM             | 50    | $84.7_{\scriptstyle \pm.0}$ | $93.6_{\scriptstyle \pm.2}$ | $46.6_{\scriptstyle \pm.3}$ | $10.5_{\scriptstyle \pm.1}$ |
|                 | 100   | $85.2_{\scriptstyle \pm.1}$ | $93.8_{\scriptstyle \pm.3}$ | $56.2_{\scriptstyle \pm.4}$ | $8.0_{\scriptstyle \pm.1}$  |
|                 | 250   | $85.4_{\scriptstyle \pm.0}$ | $94.2_{\scriptstyle \pm.1}$ | $61.4_{\scriptstyle \pm.6}$ | $6.7_{\scriptstyle \pm.1}$  |
|                 | 500   | $85.4_{\scriptstyle \pm.0}$ | $94.3_{\scriptstyle \pm.2}$ | $63.4_{\scriptstyle \pm.1}$ | $6.4_{\scriptstyle \pm.1}$  |
|                 | 1000  | $85.3_{\scriptstyle \pm.1}$ | $94.4_{\scriptstyle \pm.1}$ | $64.2_{\scriptstyle \pm.6}$ | $6.2_{\scriptstyle \pm.0}$  |
| END  | 50    | $87.1_{\scriptstyle \pm.1}$ | $84.6_{\scriptstyle \pm.5}$ | $68.6_{\scriptstyle \pm.4}$ | $5.9_{\scriptstyle \pm.1}$  |
|                 | 100   | $87.2_{\scriptstyle \pm.1}$ | $87.0_{\scriptstyle \pm.2}$ | $76.7_{\scriptstyle \pm.5}$ | $4.5_{\scriptstyle \pm.1}$  |
|                 | 250   | $87.1_{\scriptstyle \pm.1}$ | $88.5_{\scriptstyle \pm.2}$ | $80.7_{\scriptstyle \pm.6}$ | $3.5_{\scriptstyle \pm.0}$  |
|                 | 500   | $87.0_{\scriptstyle \pm.0}$ | $88.8_{\scriptstyle \pm.3}$ | $81.7_{\scriptstyle \pm.4}$ | $3.3_{\scriptstyle \pm.0}$  |
|                 | 1000  | $87.0_{\scriptstyle \pm.0}$ | $89.2_{\scriptstyle \pm.3}$ | $82.5_{\scriptstyle \pm.3}$ | $3.0_{\scriptstyle \pm.0}$  |
| **END (mean only)** | 50    | $85.6_{\scriptstyle \pm.1}$ | $87.8_{\scriptstyle \pm.2}$ | $66.0_{\scriptstyle \pm.4}$ | $7.9_{\scriptstyle \pm.0}$  |
|                 | 100   | $85.8_{\scriptstyle \pm.1}$ | $89.9_{\scriptstyle \pm.1}$ | $73.7_{\scriptstyle \pm.4}$ | $6.1_{\scriptstyle \pm.1}$  |
|                 | 250   | $85.7_{\scriptstyle \pm.1}$ | $91.2_{\scriptstyle \pm.2}$ | $77.4_{\scriptstyle \pm.4}$ | $5.0_{\scriptstyle \pm.1}$  |
|                 | 500   | $85.8_{\scriptstyle \pm.1}$ | $91.6_{\scriptstyle \pm.1}$ | $78.6_{\scriptstyle \pm.3}$ | $4.8_{\scriptstyle \pm.1}$  |
|                 | 1000  | $85.8_{\scriptstyle \pm.1}$ | $91.8_{\scriptstyle \pm.1}$ | $79.4_{\scriptstyle \pm.4}$ | $4.6_{\scriptstyle \pm.0}$|
We thank the reviewer for engaging in the discussion, and helping us improve the paper further. 

First, we want to reiterate that a comparison with GeoBFN based on the published results is provided in our rebuttals, and will be included in the final version of the paper — details are provided below for reference.

Second, we want to emphasize that END showcases strong performance against SOTA methods, and specifically against GeoBFN.
On QM9, END demonstrates a level of performance very similar to that of GeoBFN (see below). 
On GEOM-Drugs, END clearly outperforms GeoBFN in terms of atom stability (e.g. END’s 50-step atom stability is better than GeoBFN’s 1000-step), while GeoBFN does slightly better in terms of validity. We note that GeoLDM, a very relevant baseline, is shown to lead higher validity than both GeoBFN and END, but that connectivity and geometry quality is subpar compared to END — highlighting that concluding anything from that metric alone is difficult. 

Finally, as suggested by the reviewer, we will run and collect additional metrics for GeoBFN, and include them in the final version of the paper. However, due to the unavailability of public checkpoints, we can unfortunately not perform that experiment before the discussion period ends.




**QM9**
| Metrics / Steps |        | 50              | 100            | 500            | 1000            |
|-----------------|--------|-----------------|----------------|----------------|-----------------|
| At. Sta.        | END    | $98.6 \pm 0.0$  | $98.8 \pm 0.0$ | $98.9 \pm 0.0$ | $98.9 \pm 0.0$  |
|                 | GeoBFN | $98.28\pm 0.1$  | $98.64\pm 0.1$ | $98.78\pm 0.8$ | $99.08\pm 0.06$ |
|                 |        |                 |                |                |                 |
| Mol. Sta.       | END    | $84.6 \pm 0.1$  | $87.4 \pm 0.2$ | $88.8 \pm 0.4$ | $89.1 \pm 0.1$  |
|                 | GeoBFN | $85.11 \pm 0.5$ | $87.21\pm 0.3$ | $88.42\pm 0.2$ | $90.87\pm 0.2$  |
|                 |        |                 |                |                |                 |
| V               | END    | $92.7\pm 0.1$   | $94.1\pm 0.0$  | $94.8\pm 0.2$  | $94.8\pm 0.1$   |
|                 | GeoBFN | $92.27\pm 0.4$  | $93.03\pm 0.3$ | $93.35\pm 0.2$ | $95.31\pm 0.1$  |
|                 |        |                 |                |                |                 |
| V x U           | END    | $91.4\pm 0.1$   | $92.3\pm 0.2$  | $92.8\pm 0.2$  | $92.6\pm 0.2$   |
|                 | GeoBFN | $90.72\pm 0.3$  | $91.53\pm 0.3$ | $91.78\pm 0.2$ | $92.96\pm 0.1$  |



**GEOM-Drugs**
| Metrics / Steps |        | 50             | 100            | 500            | 1000           |
|-----------------|--------|----------------|----------------|----------------|----------------|
| At. Sta.        | END    | $87.1 \pm 0.1$ | $87.2 \pm 0.1$ | $87.0 \pm 0.0$ | $87.0 \pm 0.0$ |
|                 | GeoBFN | $75.11$        | $78.89$        | $81.39$        | $85.60$        |
|                 |        |                |                |                |                |
| V               | END    | $84.6 \pm 0.5$ | $87.0 \pm 0.2$ | $88.8 \pm 0.3$ | $89.2 \pm 0.3$ |
|                 | GeoBFN | $91.66$        | $93.05$        | $93.47$        | $92.08$        |
Thank you for the response. However I do believe the paper would be in better shape by additional efforts in some modifications e.g., adding comparison with advanced baselines (e.g., GeoBFN). I understand that GeoBFN was published ~2months prior to the deadline but the whole experiment would just take around several days and a fair comparison is expected, since you both work on the same dataset and benchmark and claim the advantage of fewer step sampling. Moreover, the work still seems technically incremental. I will slightly increase the score but still hold an opinion that the paper could be further strengthened by either showcasing strong performance against sota methods or exploring novel applications/use cases that constitutes a unique contribution.
Again, we want to thank the reviewer for providing constructive feedback. 

We believe that we have now addressed the weaknesses raised in the initial review. 

We are happy to clarify any issue that should remain.
We want to thank the reviewer for re-evaluating our submission, and providing a positively updated score. 
We will make sure to include all rebuttal updates in the final version of the manuscript.
I appreciate authors' efforts in addressing my concerns and questions in rebuttal. After reading over authors' rebuttal responses and pdf, I think all my concerns have been addressed so I increased my score. I hope authors will carefully add all rebuttal updates (discussion, analysis and experiment results) to the revised version of paper in the future.