Diffusion-based Molecule Generation with Informative Prior Bridges
Authors: Lemeng Wu
Chengyue Gong
Xingchao Liu
Mao Ye
qiang liu
Venue: NeurIPS 2022 Accept (2022)

Abstract: AI-based molecule generation provides a promising approach to a large area of biomedical sciences and engineering, such as antibody design, hydrolase engineering, or vaccine development. Because the molecules are governed by physical laws, a key challenge is to incorporate prior information into the training procedure to generate high-quality and realistic molecules. We propose a simple and novel approach to steer the training of diffusion-based generative models with physical and statistics prior information. This is achieved by constructing physically informed diffusion bridges, stochastic processes that guarantee to yield a given observation at the fixed terminal time. We develop a Lyapunov function based method to construct and determine bridges, and propose a number of proposals of informative prior bridges for both high-quality molecule generation and uniformity-promoted 3D point cloud generation. With comprehensive experiments, we show that our method provides a powerful approach to the 3D generation task, yielding molecule structures with better quality and stability scores and more uniformly distributed point clouds of high qualities. 

Diffusion-based Molecule Generation with Informative Prior Bridges
Lemeng Wu"
Chengyue Gong" University of Texas at Austin cygongōes. utexas. edu
University of Texas at Austin lmwudcs. utexas.ede
Xingchao Liu University of Texas at Austin xcliuêcs. atexas.edu
Mao Ye
Qiang Liu
University of Texas at Austin my210cs utexas. edu
University of Texas at Austin lqianggcs utexas. edu
Abstract
Al-based molecule generation provides promising approach to large area of biomedical sciences and engineering, such antibody design, hydrolase engineer- ing, or vaceine development. Because the molecules are governed by physical laws, key challenge is to incorporate prior information into the training procedure generate high-quality and realistic molecules. We propose a simple and novel ap- proach to steer the training of diffusion-based generative models with physical and statistics prior information. This is achieved by constructing physically informed diffusion bridges, stochastic processes that guarantee yield given observation at the fixed terminal time. We develop Lyapunov function based method to construct and determine bridges, and propose number of proposals of informative prior bridges for both high-quality molecule generation and 3D point cloud generation. With comprehensive experiments, we show that our method provides powerful approach to the 3D generation task, yielding molecule structures with better quality and stability scores and more uniformly distributed point clouds of high qualities.
Introduction
As exemplified by the success of AlphafoldV2 [22] in solving protein folding, deep learning tech- niques have been creating new frontiers on molecular sciences [46]. particular, the problem of building deep generative models for molecule design has attracted increasing interest with magnitude of applications in physics, chemistry, and drug discovery [e.g., 2, 3, 26J. Recently, diffusion- based generative model have been applied to molecule generation problems [9, 19] and obtain superior performance. The idea of these methods to corrupt the data with diffusion noise and learn a neural diffusion model to revert the corruption process to generate meaningful data from noise.
key challenge in deep generative models for molecule and 3D point generation is to efficiently incorporate strong prior information reflect the physical and problem-dependent statistical prop- erties of the problems at hand. In fact, a recent fruitful line of research [11, 1, 23, 37] have shown promising results by introducing inductive bias into the design of model architectures to reflect physical constraints such as SE(3) equivariance. diffusion-based generative models, and leverage yield substantial
this work, we present a different paradigm of
prior incorporation tailored improvement in both 1) high-quality and stable molecule generation and 2) cefformmty--romoote point cloud generation. Our contributions summarized follows.
Equal contribution
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Prior Guided Learning of Diffusion Models. We introduce a simple and flexible framework for injecting informative
prior and physical information when learning diffusion- based generative models. The idea is to elicit and inject prior information regarding how the diffusion process should look like for generating each given data point, and train the neural diffusion model to imitate the prior processes. The prior information is presented in the form of diffusion bridges which are diffusion processes that are guaranteed to generate each data point at the fixed terminal time, We provide a general Lyapunov approach for constructing and determining bridges and leverage it to develop a way to systematically incorporate prior information into bridge processes.
Physics-informed informed Molecule Generation. We apply our method to molecule generation. We propose number of energy functions for incorporating physical and statistical prior information. Compared with existing thysics-informer molecule generation methods [e.g.. 9, 14, 29, 14], our method modifies the training process, rather than imposing constraints on the model architecture. Experiments show that our method achieves current state-of-the-an generation quality and stability on multiple test. benchmarks of molecule generation.
cniformmity-roomottie Point Generation. A challenging task in physical simulation, graphics, 3D vision to generate point clouds for representing real objects [e.g., 1, 5, 28, 51]. A largely overlooked problem of existing approaches is that they tend to generate unevenly distributed points, which lead to unrealistic shapes and make the subsequent processing and applications. such as mesh generation, challenging and inefficient. In this work, we leverage our framework to introduce uniformity- prommoiig forces into the prior bridge of diffusion generative models. This yields simple and efficient approach to generating regular and realistic point clouds in terms of both shape and point distribution.
2 Related works
Diffuse Bridge Process. Diffusion-based generative models [18, 40, 41, 44, 25] have achieved great successes in various Al generation tasks recently; these methods leverage time reversion technique and can be viewed as learning variants auto-encoders with diffusion processes as encoders and decoders. Schrodinger bridges [7, 9, 45] have also been proposed for learning diffusion generative models that guarantee to output desirable outputs finite time interval, but these methods involve iterative proportional fittings and are computationally costly. Our framework of learning generative models with diffusion bridges similar to that of [35], which learn diffusion models as mixture of forward-time diffusion bridges to avoid the time-reversal technique of [43]. But our framework is designed to incorporate physical prior into bridges and develop a systematic approach for constructing a broad class of prior-informed bridges.
3D Molecule Generation. Generating molecule in 3D space has been gaining increasing interest. A line of works [e.g. 30, 32. 38, 39, 48, 49, 50] consider conditional conformal generation. which takes the 2D SMILE structure as conditional input and generate the 3D molecule conformations condition on the input. Another series of works [e.g., 13, 19, 27, 37,47] focus on directly generating the atom position and type for the molecule unconditionally. For these series of works, improvements usually come from architecture design and loss design. For example, G-Schnet [13] auto- regreseegrssively generates the atom position and type one by one after another; EN-Flow [37] and EDM [19] adopt E(n) equivariant graph neural network (EGNN) [37] to train blow- model and diffusion model. These methods aim at generating valid and natural molecules in 3D space and outperform previous approaches by a large margin. Our work provides very different approach to incorporating the physical information for molecule generation by injecting the prior information into the diffusion process, rather than neural network architectures.
Point Cloud Generation. vast literature has been devoted learning deep generative models for real-world 3D objects in the form of point clouds. [1] first proposed to generate the point cloud by generating latent code and training a decoder to generate point clouds from the latent code. Build upon this approach, methods have been developed using flow-based generative models [51] and diffusion-base models [5, 28, 29]. However, the existing works miss a key important prior information: the points in point cloud tend to distribute regularly and uniformly. Ignoring this information causes poor generation quality. By introducing mniformity-promotiin forces in diftusion bridges, obtain simple and efficient approach to generating regular and realistic point clouds.

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3 Method
We first introduce the definition diffusion generative models and discuss how to learn these models with prior bridges. After introducing the training algorithm for deep diffusion generative models, we discuss the energy functions that we apply to molecules and point cloud examples.
3.1 Learning Diffusion Generative Models with Prior Bridges
Problem Definition. We aim at learning generative model given a dataset {x(4) Jk=1 drawn from an unknown distribution II on IRd. A diffusion model on time interval [0, 1]] is
p az, o (Z,)dt + (Z,)dWW Vt E [0,1], Zu ~ plo
where W. is a standard Brownian motion; oz: Rd + Rdxd is positive definition covariance coefficient; sy: Rd Rd is parameterized as neural network with parameter 0, and fio is the initialization. Here we use po to denote the distribution of the whole trajectory Z {z, E [0, I]]. and P the marginal distribution of Z, time t. We want to learn the parameter 0 such that the distribution PP of the terminal state Z1 equals the data distribution 11".
Learning Diffusion Models. There are an infinite number of diffusion processes I that yield the same terminal distribution but have different distributions of latent trajectories Z. Hence, it is important to inject problem-dependen prior information into the learning procedure to obtain a model p that simulate the data for the problem hand fast and accurately. To achieve this, we elicit imputation process Q for each x E Rd, such that a draw Z ~ Q yields trajectories that 1) are consistent with x: in that Z deterministically, and 2) reflect important physical and statistical prior information on the problem at hand.
Formally, if Q* x) 1. call that Q is bridge process pinned end point simply an z-bridge. Assume we first generate a data point x ~ II", and then draw a bridge Q pinned x, then the distribution of Z is mixture of Q" with drawn from the data distribution: Q"" f (-II" (dx).
A key property of Q"" is that its terminal distribution equals the data distribution, i.e., Q = II" Therefore, we can learn the diffusion model p by fitting the trajectories drawn from Q with the "backward" procedure above. This can be formulated by maximum likelihood or equivalently minimizing the KL divergence:
min = =CC)"""((""" || pP))
Furthermore, assume that the bridge Q3 is diffusion model of form
(1) where (2, x) is an x-dependent drift term need to carefully designed to both satisfy the bridge condition and incorporate important prior information (see Section 3.2). Assuming this is done, using Girsanov theorem [33], the loss function C(0) can be reformed into form ofdenoised score matching
Q" dz, = (2, x)dt + (Z))dW, Zo ~ po.
loss of [e.g., 41, 43, 42]:
2 /
[()) E Qm
[o(Z))
(,
2 dt + const,
which is score matching term between 8 and b. The const term contains the logelikelihood for the initial distribution po, which is a const in our problem. Here #* is an global optimum of L(0) if
L (²) Ez-gge (b)(2] Z, = =].
This means that the drift term of should matched with the conditional expectation b,(2)z) with Zi conditioned on Z, z.
Remark 3.1. The SMLD can be viewed as a special case of this framework when we take Q to be a time-scaled Brownian bridge process:
2 I 7, dt + o,WW, Zo (x,
Db.bb
dZ,
0, 31 8,
(3)
3

---

where E [0, +20) and B, = dz This can be seen by the fact that the time-reversed process Ze -1 the simple time-scaled Brownian motion dZ, -w_, starting from the data point Zo z. where Wt is another standard Brownian motion. The Brownian bridge achieves Z1 = z because the magnitude of the drift force is increasing to infinite when is close to time 1.
However, the bridge of SMLD above is relative simple and uninformative process and does not incorporate wobblemddependent prior information into the learning procedure. This is also the case of the other standard diffusion-
models [43], such as denoising diffusion probabilistic models (DDPM) which can be shown to use a bridge constructed from an Ornstein-Uhlenbeck process. We refer the readers to [35], which provides similar forward time bridge framework for learning diffusion models, and it recovers the bridges in SMLD and DDPM as a conditioned stochastic process derived using the h-transform technique [10]. However, the h-transform method is limited to elementary stochastic processes that have an explicit formula of the transition probabilities, and can not incorporate complex physical statistical prior information. Our work strikes construct and use broader class of more complex bridge processes that both reflect problem-dependent prior knowledge and satisfy the endpoint condition Q (Z1 x) 1. This necessitate systematic techniques for constructing a large family of bridges, as we pursuit in Section 3.2.
3.2 Designing Informative Prior Bridges
The key to realizing the general prior- informnn learning framework above is to have general and user-friendly technique to design Q in (1) to ensure the bridge condition Q² = x) = while leaving the flexibility of incorporating rich prior information. To achieve this, we first develop general criterion of bridges based on a Lyapunov function method which allows us to identify very general form of bridge processes; we then propose particularly simple family of bridges that we use in practice by introducing modification to Brownian bridges.
Definition 3.2 (Lyapunov Functions). A function Uz(z) is said to be a Lyapunov function for set C Rd at time t = if (x) 2 0 for V2 IRd and Ui (2) if and only if z E A.
Intuitively, diffusion process Q is bridge A, i.e., QZz A) 1, if it (at least) approximately follows the gradient flow of a Lyapunov function and the magnitude (or step size) or the gradient flow should increase with a proper magnitude in order to ensure that Z, A at the terminal time 1. Therefore, we identify a general form of bridges to A as follows:
d2, = (-a,V, + (2,) dt + (Z.)dWW.
t E [0, Zo no:
(4)
where ar > 0 is the step size of the gradient flow of U and v is an extra perturbation term. The step size a₂ should increase to infinity as 1 sufficiently fast to dominate the effect of the diffusion term o,WW, and the perturbation dedt term to ensure that U is minimized at time t = 1.
Proposition 3.3, Assume U²(=)= = U(2, t) is Lyapunov function of a measurable set A time and U-.t) E (RR) and (2,-)) E C* ([O, 1)). Then, QA in (4) an bridge to A, iee. QA E A) = if the following holds:
1) U follows an (expected) Polyak- Lojjsiwwoj condition:
- [Z) - s0
2) Let Be = B.al (Z))), and 7: = (Z, + t [1((V())(.))) and exp(See a,ds). Then limeç G +00, and lim Jg (8. +v.
6
+
Brownian bridge can be viewed as the case when (z( = || 211 /2 and of (31 - B.). and Henee simply introducing an extra drift term into bridge bridge yields that a broad family of bridges to x:
dz, (orf((Z))+ 0f 2- Zz
B1 3, dt Hod,,, Zu ~ po-
(5)
In Appendix A.4 and A.5, we show that Dr.
ff is a bridge to x if [[] < +oc and o. 0, Vt. which is very mild condition and is satisfied for most practical functions. The intuition is that the Brownian drift of x- z-Z, is singular and grows to infinite as approaches 1. Hence,
B.

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GNN
Drift
Char Coor
D.5
x(t)
Noise
Energy
Figure overview of our training pipeline with molecule generation as an example. Initialized given distribution, we pass the data through the network multiple times, and finally get the meaningful output.
introducing an into the drift would not change of the final bridge condition, unless f is also singular and has magnitude that dominates the Brownian bridge drift as t
To make the model pe compatible with the physical force f. we assume the learnable drift has form of (=) a=)(=) s8 (z) where & is neural network (typically a GNN) and can be another learnable parameter or pre-defined parameter. Please refer to algorithm 3.2 and Figure for descriptions about our practical algorithm.
Algorithm Learning diffusion generative models.
Input: Given dataset (x(*k), Q" the bridge (5), and a problem-dependeed prior force f.v and diffusion model Iº
Training: Estimate 0 by minimizing [(0) in (2) with stochastic gradient descent and time dis- cretization.
Sampling:
Simulate from pd.
4 Molecule and 3D Generation with Informative Prior Bridges
We apply our method the molecule generation as well as point cloud generation. Informative physical or statistical priors that reflects the underlying real physical structures can be particularly beneficial for molecule generation we show in experiments.
In our problem, each data point æ is collection of atoms of different types, more generally marked points, in 3D Euclidean space. In particular, coordinate of the i-th atom, and x {ei....,ey' ex) where each e; (0... .......... 1... 0] is the i-th basis vector of RR which indicates the type of the i-th atom of A categories. To apply the diffusion generative model, we treat x. as continuous vector R" and round it to the closest basis vector when want to output final result or have computations that depend on atom types (e.g., calculating an energy function as we do in sequel). Specifically, for a continuous x E IR we denote by a I( max(2?)) the discrete type rounded from by taking the type with the maximum value. To incorporate priors, we design an energy function E(x) and incorporate fi(-) - -VE() into the Brownian bridge (5) guide the training process. We discuss different choices of E the following.
have
[ x m where xt R is the
4.1 Prior Bridges for Molecule Generation
Previous prior guided molecule protein 3D structure generation usually depends on pre-defined energy or force [30, 50]. We introduce our two potential energies. One is formulated inspired by previous works in biology, and the other is an nearest neighbour statisties directly obtained from the data.
AMBER Inspired Physical Energy. AMBER [12] is family of force fields for molecule sim- ulation. is designed to provide a computationally efficient tool for modern chemistry-mon molccular dynamics and free energy calculations. It consists of number of important forces, including the bond energy, angular energy, torsional energy, the van der Waals energy and the Coulomb energy. Inspired by AMBER, propose to incorporate the following energy term into the bridge process:
E(x) = (x))d(x)) + Eangte (x) + EI,(x) + (Coulomb(z))
5

---

The bond energy is (x)nd(0) Eijebond( (Len(xl,) E(0f, 25))), where Len(x.y) ||: - -jil.- and bond(x) denotes the set bonds from x, which is set to be the set of atom pairs with a distance smaller than 1.15 times the covalent radius; the (r, c) denotes the expected bond length between atom type and c, which we calculate as side information from the training data.
The angle energy is Eangle (z) = E Zijkgangte(a) (Ang(r)5)) - ( (d))))) where angle(x) denotes the set of angles between two neighbour bonds in bo)nd((r), and Ang( denotes the angle formed by vector x. z and 2 2j. and ( ) is the expected angle between atoms of type at, 4/. ggg. which we calculate as side information from the training data.
The Lennard-Jones (LJ) energy is defined by (z)(2) - Eizf - and e(t) (0/8)22- 2() The parameter a is an approximation for average nucleus distance.
The nuclei-nuclei repulsion (Coulomb) electromagnetic potential energy is ECoutomb (x) Eis (( (g (2))/ |2] zjil where is Coulomb constant and q(r) denotes the point charge of atom of type r, which depends on the number of protons.
Statistical Energy. When accurate physic laws unavailable, molecular geometric statistics, such as bond lengths, bond angles, and torsional angles, etc, can be directly calculated from the data and shed important insights on the system [e.g., 8, 21, 3 1]. We propose to design prior energy function in bridges by directly calculate these statistics over the dataset.
Specifically, we assume that the lengths and angles of each type of bond follows Gaussian distribu- tion that we learn from the dataset, and define the energy function as the negative ligeliooihood::
Estat (I) =
Σ
Len(xly) - Patt || +
Σ
Ang(E)))) - Mat l| (7)
||
()E
(.jkkknn
where kn)((r) denotes the K-nearest neighborhood graph constructed based on the distance matrix of a; for each pair of atom types r, e E [k), Pre and afc denotes empirical mean and variance of length of re-edges in the dataset; for each triplet C, E Aver" and 2² the empirical mean and variance of angle betwen re and cr bonds.
Intuitively, depending on the atom type and order of the nearest neighbour, we force the atom distance and angle to mimic the statistics calculated from the data. We thus implicitly capture different kinds of interaction forces. Compared with the AMBER energy, the statistical energy (7) is simpler and more udaptive to the dataset of interest.
4.2 Prior Bridges for Point Cloud Generation
We design prior forces for 3D point cloud generation, which is similar molecule generation except that the points are un-typed so we only have the coordinates } One important aspect of point cloud generation is distribute points uniformly on the surface, which is important for producing high-quality meshes and other post-hoc geometry applications and manipulations.
Riesz Energy. One idea to make the point distribute uniformly is adding a repulsive force to separate the points apart from each other [24, 15, 16]. We achieve this by minimizing the Riesz energy [17],
E (x)
Σ - j
(8)
2
KNN Distance Energy. Similar molecule design, directly calculate the average distance between each point and its k nearest neighbour neighbour, and define the following energy:
Exan (x) Σ (knn-dist; (kkn)
where knn-dist, E)eN | 12 denotes the average distance from T; to its K nearest neighbors, and joen is the empirical mean of knn-dist,(x) the dataset. This would encourage the points to have similar average nearest neighbor distance and yield uniform distribution between points. In common geometric setups, the valence of the point on the surface is 4, which means we set l 4.

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Experiment
We verify the advantages of our proposed method (Bridge with Priors) in several different domains. We first compare our method with advanced generators (e.g., diffusion model, normalizing flow, etc.) on molecule generation tasks. We then implement our method on point cloud generations, which targets producing generated samples in higher quality. We directly compare the performance and also analyze the difference between our energy prior and other energies we discuss in Section 3.
5.1 Force Guided Molecule Generation
To demonstrate the efficiency and effectiveness of our bridge processes and physical energy, we conduct experiments on molecule and macro-molecule generation experiments. We follow [29] in settings and observe that our proposed prior bridge processes consistently improve the state-of-the-art performance. Diving deeper, we analyze the impact of different energy terms and hyperparameters.
Metrics. Following [19, 37]. we use the atom and molecular stability score to measure the model performance. The atom stability the proportion of atoms that have the right valency while the molecular stability stands for the proportion of generated molecules for which all atoms are stable. For visualization, types, which is a common practice. we extracted 10,000 samples to calculate the above metrics.
the distance between pairs of atoms and the atom types predict bond
Dataset Settings QM9 [36] molecular properties and atom coordinates for 130k small molecules with up to 9 heavy atoms with 5 different types of atoms. This data set contains small amino acids, such GLY, ALA, well nucleobases cytosine, uracil, and thymine. We follow the common practice in [19] to split the train, validation, and test partitions, with 100K, 18K, and 13K samples. GEOM-DRUG [4] is dataset that contains drug-like molecules. It features 37 million molecular conformations annotated by energy and statistical weight for over 450,000 molecules. Each molecule contains 44 atoms on average, with different types of atoms. Following [19, 37], retain the 30 lowest energy conformations for each molecule.
Training Contigurations. On QM9, train the EGNNs with 256 hidden features and 9 layers for 1100 epochs, a batch size 64, and a constant learning rate 10 which is the default training configuration. We use the polynomial noise schedule used in [19] which linearly decay from 10-2 //T to O. We linearly decay a from 10-3 /T to O w.E.f. time step. We set & 5 (7) by default. On GEOM-DRUG, we train the EGNNs with 256 hidden features and 8 layers with batch size 64, constant learning rate 10 and 10 epochs. It takes approximately 10 days to train the model on these datasets
Tesla V100-SXM2-333 SXCB-322B GPU. We provide H(3) Equivariant Diffusion Model (EDM) [19] and E(3) Equivariant Normalizing Flow (EN-Flow) [37] as our baselines. Both two are trained with the same configurations ours.
Table 1:
Results of our method and several haselines QM9 and GEOM-DRUG. We evaluate the percentage of valid and unique molecules out of 12000 generated molecules.
QM9
GROM-DRUG
Atom Sta (%)
Mol Sta (
Valid Unique
Atom Sta (%)
Mol Sta (%)
EN-Flow [37]
85.0
4.9
0.349
75.0
0.0
GDM [19]
97.0
63.2
75.0
0.0
E-GDM [19]
98.7::0.1
82.0=04
0.902
81.3
0,0
Bridge
98.7:10.1
$1.810.2
0.902
81.020.7
0.0
Bridge Force (7)
98.8:10.1
84.6=0.3
0.907
82.4=0.8
0.0
Q.99
GEOM-DRUG
Figure 2:
Examples of molecules generated by our methad on QM9 and GEOM-DRUG.
observe that (1) our method generates molecules with better qualities than the others. On QM9, we notice that we improve the molecule stability score by large margin (from 82.0 to 84.6) and slightly improve the atom stability score (from 98.7 to 98.8). It indicates that with the informed prior bridge helps improves the quality of the
Results. We summarize our experimental results Table
7

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Time Step
Figure 3: An example of generation trajectory following Io of our method, trained on HEOM-DRUGG
Table 2: We compare and w/o force results with different discretization time steps.
Time
50
100
500
Atom Stable (%)
Mol Stable (%)
Atom Stable (%)
Stable (%)
Atom Stable (S)
(%)
EGM
97.0=0,1
66.4=0.22
97.3=0,1
69.8+0.2
98.5=0.1
81.2=0,1
Bridge + Force (7)
9..3
69.2:10.2
97.9.10.1
72.3:10.2
98.7-001
83.7:00.1
generated molecules. (2) On the GEOM-DRUG dataset, the atom stability is improved from 81.3 to 82.4, which shows that our method can work for macro-molecules. (3) We visualize and qualitatively evaluate our generate molecules. Figure 3 displays the trajectory on GEOM-DRUG and Figure 2 shows the samples on two datasets. (4) Bridge processes and E-GDM obtain comparable results on our tested benchmarks. (5) The computational load added by introducing prior bridges is small. Compared to EGM, we only introduce 8% additional cost in training and 3% for inference.
Result: Better With Fewer Time Steps. We display the performance of our method with fewer time steps in Table We observe that (1) with fewer time steps, the baseline EGM method gets worse results than 1000 steps in Table 1. (2) with 500 steps, our method still keeps consistently good performance. (3) with even fewer 50 or 100 steps, our method yields worse result than 1000 steps in Table 1, but still outperforms the baseline method by a large margin.
Table 3:
We compare EGM models trained with different force mentioned in Section 3.
Method
Atom Stable (%)
Mol Stable (%)
Method
Atom Stable (%)
Mol Stable (%)
Force (7), k =
98.8:10.1
84,5::0.2
Foree (6)
98.7
83.1=0.2
Force (7), k
98.8++0.1
84.6++0.3
Force (6) w/o. bond
98.7+0.11
82.5+0011
Foree (7). k
98.8:10.1
83.910.3
Force (6) angle
98.7100.1
82.420.2
Foree (7), k
98.8-+0.11
82,7++0.3
Force (6) w/o. Long-range
98.7+00.1
82.7+002
Ablation: Impacts of Different Energies. We apply several energies we discuss in Section 3, and compare them the QM9 dataset. (1) We notice that our energy (7) gets better performance with larger k when k < 5. k = achieves comparable performance more computation time, which yields a trade-off between performance and efficiency. (2) For (6), once removing a typical term, the performance drops. (3) In all the cases, applying additional forces outperforms the bridge processes baseline w/o. force.
k = 5. Larger is also requires
5.2 Force Guided Point Cloud Generation
We apply uniformity-promme promoting priors to point cloud generation. We apply our method based on the diffusion model for point cloud generation introduced by point cloud diffusion model [28] and compare it with the original diffusion model well the case of bridge processes w/o. force prior. We observe that our method yields better results in various evaluation metrics under different setups.
Dataset. We use the ShapeNet [6] dataset for point cloud generation. ShapeNet contains 55 categories. We select Airplane and Chair, which are the two most common categories to evaluate in recent point cloud generation works [5, 28, 51, 52]. We construct the point clouds following the setup in [28], split the train, valid and test dataset in 80%, 15% and 5% and samples 2048 points uniformly on the mesh surface.
Evaluation Metric. We evaluate the generated shape quality in two aspects following the previous works, including the minimum matching distance (MMD) and coverage score (COV). These scores are the two most common practices in the previous works. We use Chamfer Distance (CD) and Earth Mover's 's Distance (EMD) the distance metric to compute the MMD and COV.

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Experiment Setup. We train the model with two different configurations. The first one uses exactly the same experiment setup configuration introduced in [28]. Thus, we use the same model architecture and train the model in 100 diffuse steps with learning rate 2 x 10-3 3 batch size 128, and linear noise schedule from 0.02 10-1. We initial a with 0.1 and jointly learn with the network, For the second setup, to evaluate the better converge speed of our method, we decrease the diffuse step from 100 to 10 with other settings the same. For the diffusion model baseline, we reproduce the number by directly using the pre-trained model checkpoint and testing it the test set provided by the official codebase.
Diffusion
Bridge
Bridge+Riesz
Bridge+Statistic
Figure 4: From left right are examples of point clouds generated by [28]. method with uniformative
bridge (fr = 0), bridge with Riesz energy (f. -V ERisO)) and with KNN energy (fi Exnn). see that the Riesz and KNN energies yield more uniformly distributed points. Riesz energy sometimes creates additional outlier points due to its repulsive nature.
Result. We show our experimental result in Table 4. We that (1) In the 10 steps setup, all variants of our approach are clearly stronger than the diffusion model. With force added, our method with physical prior achieves nearly the same performance the 100 steps setup. (2) In the 100 steps setup, adding energy potential as prior improves the bridge process performance and further let beat the diffusion model baseline. Since the test points uniformly sampled on the surface, better score indicates a closer point distribution to the reference set. Further, when compare with Riesz energy (8), statistic gap energy (9) performs better. One explanation is the Riesz energy pushes the points to some outlier position in sample generate samples, while statistic gap energy is more robust. We also show visualization samples in Figure 4.
Table 4:
Point cloud generation results. CD is multiplied by 10 EMD is multiplied by 10.
10 Steps
100 Steps
MMD
COV
MMD
COV +
CD
EMD
CD
EMD
CD
EMD
CD
EMD
Diffusion [28]
14.01
3.23
32.72
29.36
12.32
1.79
47.41
47.59
Bridge
13.04
2.14
46.01
42.59
12.47
1.85
47.83
47.13
Chair
Riesz
12.84
1.95
47.21
44.31
12.31
1.82
48.14
47.42
Statistic
12.65
1.84
47.58
45.23
12.25
1.78
48.39
47.56
Diffusion [28]
3.71
1.31
43.12
39.94
3.28
1.04
48.74
46.38
Bridge
3.44
1.24
46.90
43.46
3.37
1.08
47.11
46.17
Airplane
Riesz
3.39
1.20
47.11
43.12
3.24
1.09
48.62
46.23
+ Statistic
3.30
1.12
47.02
44.67
3.24
48.53
46.73
6 Conclusion and Limitations
We propose framework to inject informative priors into learning neural parameterized diffusion models, with applications to both molecules and 3D point cloud generation. Empirically, demonstrate that our method has the advantages such as better generation quality, less sampling time and casy-to-calculate potential energies. For future works, we plan to 1) study the relation between different types of forces for different domain of molecules, 2) study how to generate valid proteins in which the number of atoms is very large, and 3) apply our method more realistic applications such as antibody design or hydrolase engineering.
In both energy functions in (7) and (6), do not add torsional angle related energy [20] mainly because it is hard to verify whether four atoms are bonded together during the stochastic process. We plan to study how include this for better performance in future works.
Another weakness of deep diffusion bridge processes are their computation time. Similar to previous diffusion models [29], takes a long time to train model. We attempted to speed the training up by

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using large batch size (e.g., 512, 1024) but found performance drop. An important future direction study methods distribute and accelerate the training.
Acknowledgements Authors are supported in part by CAREER-1846421, SenSE-2037267, EAGER-2041322 and Office of Navy Research, and NSF AI Institute for Foundations of Ma- chine Learning (IFML). We would like to thank the anonymous reviewers and the area chair for their thoughtful comments and efforts towards improving our manuseript.
10

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13


Reviews:
[{"review_id": "kJKqmE6werF", "reviewer": "Reviewer_cHTF", "summary": "This paper introduces a novel diffusion-based generative model that leverages the prior information to learn the diffusion process satisfying the bridge condition. Specifically, the paper proposes a  Brownian bridge with extra drift term that incorporates the prior information, and aims to learn the diffusion process by fitting the trajectories drawn from the mixture of these bridges. The paper utilizes Lyapunov functions to guarantee the bridge condition, and further introduces energy functions to guide the training process for each downstream task. The proposed method, Bridge with Priors, is able to generate samples with better quality compared to the baselines in both molecule generation and point cloud generation tasks.", "questions": "1. Is Bridge with Priors designed for a more complex process such as Ornstein–Uhlenbeck process? Or is it applicable only to the Brownian bridge which is possible due to the singular characteristic of the drift term $\\sigma^2_t\\frac{x-Z_t}{\\beta_1-\\beta_t}$?\n\n1. How to determine the learnable parameter $\\alpha$ as in $s^{\\theta}_t(z) = \\alpha f_t(z)+\\tilde{s}^{\\theta}_t(z)$?\n\n1. What is the meaning of \"to ensure that $U$ is minimized at time $t=1$\" in line 147?\n\n1. Is the condition in line 157: $\\mathbb{E}_{\\mathbb{Q}^{x,bb}} [||f_t(Z_t)||^2] <\\infty$  correct? \nIn the supplementary file, the condition of Proposition A.4. is $\\mathbb{E} [\\int^1_0||f_t(Z_t)||^2]  < \\infty$.\n\n1. Why is the molecular stability of GEOM-DRUG 0.0 for all models? Is it because the molecules are large?\n\n1. Is the training time of Bridge with Priors larger than previous diffusion models? Additionally, although the comparison with the Schr&ouml;dinger Bridge methods [De Bortoli et al., 2021, Chen et al., 2022] is out-of-the-scope of this paper, I believe that the comparison would be interesting as Bridge with Priors may have less training/sampling time and competitive or better performance.\n\n---\n**References**\n- De Bortoli et al., Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling, NeurIPS 2021, \n- Chen et al., Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory, ICLR 2022\n", "limitations": "Training the model takes a long time, which is a common limitation of diffusion models. The paper also discusses the future works in section 6, which seems to be a promising research direction.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**\n\n1. The paper proposes a novel approach of leveraging the prior information in the form of diffusion bridges to learn the diffusion process, which is clearly different from injecting inductive bias into the model architectures. Moreover, instead of using the time-reversal technique which most of the previous works are based on, the paper learns the generation process as a mixture of diffusion bridges which was introduced in [Peluchetti, 2022], which is also a new direction for the diffusion model.\n\n1. The paper has a clear motivation of exploiting problem-dependent prior information to generate high-quality samples. Especially, for the point cloud generation, the paper addresses the problem of unevenly distributed points and provides a solution by using the Risze energy function. \n\n1. Although the paper is based on mathematically heavy theory, the paper is well-written and understandable with clear notations and formulations, except for the part explaining the mixture of diffusion bridges and the loss function (lines 102-112). \n\n1. The paper shows promising results for both molecule generation and point cloud generation tasks\n\n1. I believe the experiments are well-conducted with clear data/training/evaluation setups and compared with the state-of-the-art baselines (EDM). Especially, comparing previous diffusion models, the Bridge without Prior, and the Bridge with Prior clearly shows the advantage of exploiting prior information for the generation. \n\n**Weaknesses**\n\n1. Lines 102-111 are too concise with necessary details omitted. Without reading [Peluchetti, 2022], it would have been impossible to understand the concept of \"mixture of diffusion bridges\" denoted as $\\mathbb{Q}^{\\Pi^{\\ast}}$. \n\n1. If I understood correctly, the term $b_t(Z_t|Z_1)$ in the loss function of Eq.(2) should be the drift term of the mixture $\\mathbb{Q}^{\\Pi^{\\ast}}$, not the drift term of a specific $x$-bridge $\\mathbb{Q}^{x}$. Is this term analytically accessible? \n\n1. The computation of the loss function of equation (2) needs clarification. I assume the expectation in Eq.(2) was computed by a Monte Carlo estimate with a sample $(t,z_1,z_0,z_t)$ where $z_1\\sim \\text{Data}$, $x_0\\sim\\Pi_{0|1}(dx_0|x_1)$ and $z_t\\sim p_{t|0,1}(z_t|z_0,z_1)$. Although [Peluchetti, 2022] shows that $p_{t|0,1}(z_t|z_0,z_1)$ has explicit form, how is $x_0\\sim\\Pi_{0|1}(dx_0|x_1)$ sampled? \n\n1. The condition of line 157: $\\mathbb{E}_{\\mathbb{Q}^{x,bb}}\\|f_t(Z_t)\\|^2$ does not seem trivial which could be satisfied for most practical functions, although the provided intuition in lines 158-161 is convincing. It would be clearer if the authors provide proof that the condition is satisfied for $f$ induced from the proposed energy functions.\n\n1. The reason for incorporating the energy function (prior information) into the Brownian bridge by $f_t(\\cdot)=-\\nabla E(\\cdot)$ is not clear. Is the energy function minimized when $t$ approaches 1?\n\n1. I would like to see more experimental results to the authors' claim for \"less sampling time\", either for GEOM-DRUG or the point cloud generation tasks. I can see that in Table 2, Bridge+Force with 500 steps outperforms EGM for 1000 steps in the QM9 dataset which is impressive, but as the molecules of QM9 are small, it would be more convincing if the claim is proven for larger datasets. It would be great if there is an ablation study section in the paper to emphasize the advantage of less sampling time.\n\n(*) I would like to raise my score if the presentation is improved and the addressed concerns/questions are clarified.\n\n**Minor correction**\n- The explanation for the abbreviation SMLD is missing. \n- The SDE of $\\mathbb{P}^{\\theta}$ and Eq.(1) use t-subscript notation for drift and diffusion coefficients, while Eq.(4) does not use such t-subscript. I recommend unifying the notation to prevent confusion.\n- No bold in row Airplane, column 100 Steps-COV-EMD in Table 4.\n- Condition #3 of Theorem A.1. in the supplementary file is missing.\n- In my opinion, Algorithm 1 and Figure 1 do not give much information. It would be great to add more details about \"mixture of diffusion bridges\" and the explicit form of $b_t(Z_t|Z_1)$.\n\n---\n**References**\n- Peluchetti, Non-denoising forward-time diffusions, 2022", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "c9QB_2gGZu", "reviewer": "Reviewer_nq85", "summary": "This paper proposes a new diffusion method incorporating physical information by carefully designing the prior for the downstream task. The suggested diffusion approach incorporates physical prior into bridges, in contrast to other diffusion methods that learned diffusions as a combination of forward-time diffusion bridges. The method was evaluated on molecules and 3D point cloud generation tasks, results of which show advantages in terms of generating quality, efficiency, and energy calculation convenience. They demonstrate the superiority of the method in terms of both quantity and quality.\n", "questions": "* Taking fewer time steps sometimes could mislead because actual wall clock time spent for each time step could be different. Wondering if a wall clock comparison can be presented.\n", "limitations": "* The scientific significance of merely producing molecular conformation is limited. The usefulness of this work would be higher if this model could be applied to real-world drug discovery applications, as the authors emphasized.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strength\n* The paper is generally clear, well structured, and easy to follow.\n* The theory and accompanying research are explained clearly.\n* They properly motivate the need for the method and describe the proposed model thoroughly.\n* Injecting task-dependent priors into a diffusion method is non-trivial and novel\n* They reasonably select the benchmarks emphasizing the strength of the suggested model.\n\n\n### Weakness\n* Typos\n  * Line 42: repeated citation [14] \n  * Line 109: ofdenoised -> of denoised\n  * Line 196: bound(x) -> bond(x)\n  * Line 240: [26] seems to be an incorrect citation because it's nothing to do with a molecule application.\n* Table 1 omits uniqueness comparisons that are important to determine the superiority of the model. Including uniqueness comparisons is important because there may be models that generate molecules with low uniqueness and high novelty only. (I'd be happy to increase my score if this issue is resolved)\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "QBnQxxG-A4f", "reviewer": "Reviewer_3WV1", "summary": "This paper proposes a framework for incorporating physics driven prior bridges in diffusion models for improved molecule and point cloud generation. In order to do so, a Lyapunov function based method is developed to construct and determine bridges. The work seems to improve on existing methods for molecule and point cloud generation.", "questions": "See questions in weaknesses sections above.", "limitations": "The limitations are mentioned in the weaknesses section above.", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "***Strengths***\n1) The proposed contribution of Lyapunov function to construct prior bridges is novel in my opinion and is a sensible approach for incorporating informative priors.\n\n2) The proposed method seems to improve molecule generation and point cloud generation benchmarks.\n\n***Weaknesses***\n\n1) For GeomDrug datasets, although the atom stability seems to improve, the overall molecule stability seems to be 0%, i.e., none of the predicted molecules are correct. Although this is the case with other existing models as well, it is surprising to see that the proposed method using informative physics informed priors does not improve this aspect of generation. Will inferring the edges during the diffusion process (instead of doing so in a post processing manner with RDKit) improve this aspect?\n\n2) I did not find any mention of computational complexity of the proposed method. It would be nice to see how feasible is the proposed approach from a practical perspective. In drug discovery, often a deserted outcome is the de novo generation of molecules at a reasonable speed. Hence, knowing about the computational requirements and the time taken on average for generating realistic molecules will be a good information to have.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept