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Diffusion-based Molecule Generation with Informative Prior Bridges
Lemeng Wu, Chengyue Gong, Xingchao Liu, Mao Ye, Qiang Liu
TL;DR
Molecule generation needs ways to incorporate physical and statistical priors into diffusion-model training. The paper constructs informative diffusion bridges with a Lyapunov-based method and applies them to molecule and 3D point-cloud generation, reporting better molecule quality and stability and more uniform point clouds. The approach also has training-time and modeling limitations that motivate future work.
Problem
Diffusion-based molecule generation must incorporate physical and statistical prior information to produce high-quality, realistic molecular structures.
Method
The paper trains diffusion models with physically informed bridges guaranteed to yield each observation at a fixed terminal time, constructed using a Lyapunov-function approach.
Results
Experiments report better molecule generation quality and stability and more uniformly distributed, high-quality 3D point clouds.
Takeaways & Limitations
The framework provides a prior-injection approach for molecule and 3D point-cloud generation with better generation quality, less sampling time, and easy-to-calculate potential energies.
Takeaways & Limitations
Training takes a long time, larger batch sizes caused performance drops, and the current energy functions omit torsional-angle information.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper targets diffusion-based molecule and 3D point-cloud generation by injecting physical and statistical priors into training. It uses informative diffusion bridges to improve molecule quality and stability while promoting uniform point distributions.
- Diffusion-based molecule generation offers applications across physics, chemistry, and drug discovery, but incorporating strong prior information remains a key challenge.
- The framework trains diffusion models to imitate problem-dependent prior processes represented as bridges guaranteed to generate each data point at a fixed terminal time.A general Lyapunov-function approach constructs and determines these bridges.
- Molecule experiments achieve state-of-the-art generation quality and stability across multiple test benchmarks without modifying model architecture constraints.The method instead modifies the training process using physical and statistical energy functions.
- Uniformity-promoting bridges generate regular, realistic point clouds by incorporating forces that improve point distribution as well as shape.The approach addresses uneven point distributions that make shapes unrealistic and downstream mesh generation challenging.
2 Related works
Prior diffusion-model work spans generative diffusion and bridge processes, 3D molecule generation, and point-cloud generation. Existing approaches commonly emphasize architectures or losses while overlooking informative physical priors such as point uniformity.
- Diffuse Bridge Process: Diffusion models and Schrödinger bridges support generative modeling, but Schrödinger bridges require costly iterative proportional fitting.
- 3D Molecule Generation: 3D molecule-generation methods either condition conformer generation on 2D SMILES or directly generate atom positions and types unconditionally.Improvements in these lines usually come from architecture and loss design.
- Point Cloud Generation: Point-cloud generation methods include latent-code decoders, flow-based models, and diffusion models, but commonly miss the prior that points distribute regularly and uniformly.
3 Method
The method learns diffusion generative models from data-pinned prior bridges, whose drift incorporates domain information while guaranteeing the desired terminal endpoint. A Lyapunov-function construction provides a general bridge criterion, with practical Brownian-bridge modifications and force-guided training.
- Learning Diffusion Models: The goal is to learn a diffusion model whose terminal state follows the data distribution, using bridge trajectories generated for individual observations.A bridge Qx is pinned at x when Qx(Z1 = x) = 1, and mixtures over data points retain the data terminal distribution.
- Learning Diffusion Models: The neural model is trained by matching its score or drift to the conditional bridge drift through a denoised score-matching objective.Girsanov's theorem reformulates the likelihood objective into a score-matching loss between sθ and b.
- Learning Diffusion Models: Standard SMLD and DDPM bridges are simple and uninformative, while the proposed framework targets broader bridges that encode complex physical and statistical priors.The h-transform approach is limited to elementary processes with explicit transition probabilities.
- Designing Informative Prior Bridges: A Lyapunov-based bridge follows a function's gradient flow toward a target set, with a step size increasing sufficiently near the terminal time to dominate diffusion and perturbation.The bridge dynamics use drift −αt∇zUt(Zt) + νt(Zt) and diffusion σt(Zt)dWt.
- Designing Informative Prior Bridges: Brownian bridges arise from a squared-distance Lyapunov function, and adding an extra drift yields a broad family of bridges to a target observation.The additional drift preserves the endpoint condition under mild integrability and positivity assumptions.
- Training and Sampling: The practical algorithm supplies data, a bridge, and a prior force, minimizes the bridge-based loss with stochastic gradient descent, and samples by simulating the learned model.The implementation can use a neural network such as a GNN together with a force-compatible learnable drift.
4 Molecule and 3D Generation with Informative Prior Bridges
The framework incorporates physical and statistical priors into diffusion-based molecule and point-cloud generation through energy-guided prior bridges. It uses molecule-specific energies and uniformity-promoting forces to guide generated structures and point distributions.
- Framework: The method applies informative physical or statistical priors to molecule and point-cloud generation through energy-guided diffusion bridges.The prior forces are incorporated into the bridge process rather than imposed through model architecture constraints.
- Representation: The framework is applied to 3D molecules represented as marked points, with atom types encoded as basis vectors and rounded to discrete types for outputs and energy calculations.Continuous atom-type vectors are rounded to the basis vector with maximum value when discrete types are required.
- Molecule generation: Molecule generation uses AMBER-inspired physical energies, including bond, angular, van der Waals, and Coulomb-related terms.The AMBER-inspired formulation uses bond lengths and angles from training data as side information, alongside Lennard-Jones and Coulomb energies.
- Molecule generation: Statistical energy models bond lengths and angles from dataset-derived empirical distributions, adapting the prior to the dataset of interest.The method assumes Gaussian distributions for bond lengths and angles and defines energy through negative log-likelihood.
- Point-cloud generation: Point-cloud generation adds repulsive and nearest-neighbor distance energies to encourage uniformly distributed points on surfaces.The KNN formulation matches average nearest-neighbor distances to their empirical mean, with k = 4 used in common geometric setups.
5 Experiment
Experiments evaluate prior-bridge diffusion models for molecule and point-cloud generation using stability, novelty, distributional, and efficiency measures. The method improves molecule quality and performs strongly with fewer diffusion steps, while energy-based priors improve point-cloud generation and uniformity.
- Experimental scope: The experiments compare Bridge with Priors against diffusion, normalizing-flow, and other baselines on molecule and point-cloud generation tasks.Evaluations include QM9, GEOM-DRUG, and ShapeNet point clouds, with analyses of energy terms and hyperparameters.
- Force Guided Molecule Generation: Molecule quality improves on QM9: molecular stability rises from 82.0 to 84.6, atom stability from 98.7 to 98.8, and novelty from 65.7 to 68.8 versus E-GDM.On GEOM-DRUG, atom stability increases from 81.3 to 82.4; the paper also notes that high novelty can reflect low-quality out-of-distribution samples.
- Force Guided Molecule Generation: With 50 or 100 diffusion steps, the method performs worse than its 1000-step result but still outperforms the baseline by a large margin.At 500 steps, it maintains consistently good performance, while the baseline worsens with fewer steps.
- Force Guided Molecule Generation: Applying additional forces consistently outperforms the bridge-process baseline without force, while larger k improves performance up to k = 5 before k = 7 becomes comparable.Larger k increases computation time, creating a performance-efficiency trade-off; removing a typical energy term reduces performance.
- Force Guided Point Cloud Generation: Riesz and KNN energies produce more uniformly distributed point clouds, but Riesz energy can create additional outlier points because of its repulsive nature.The point-cloud experiments use MMD and coverage with Chamfer Distance and Earth Mover’s Distance as distance metrics.
- Force Guided Point Cloud Generation: For point clouds, energy-prior variants outperform the diffusion baseline across evaluation metrics, with 10-step force-based models nearly matching the 100-step setup.In the 100-step setup, energy potentials improve bridge-process performance beyond the diffusion baseline; statistic gap energy performs better than Riesz energy in the reported comparison.
6 Conclusion and Limitations
The paper proposes informative priors for diffusion models applied to molecule and 3D point-cloud generation. It reports improved generation quality, reduced sampling time, and limitations involving torsional energies and computational cost.
- The framework injects informative priors into neural diffusion models for molecule and 3D point-cloud generation.
- The method achieves better generation quality, less sampling time, and easy-to-calculate potential energies.
- Torsional-angle energy is omitted because verifying whether four atoms are bonded during the stochastic process is difficult.
- Deep diffusion bridge processes require long training times, and large batch sizes caused a performance drop in attempted acceleration.
A Proofs
The appendix establishes bridge conditions using Lyapunov-function arguments and shows that suitable drift changes preserve bridge events under integrability assumptions.
- A nonnegative Lyapunov function that vanishes on the target set can establish almost-sure arrival at that set at the terminal time.The proof uses Ito's lemma and Grönwall's inequality to show the terminal Lyapunov value is zero.
- The Lyapunov analysis requires bounds expressed through sequences α_t, β_t, and γ_t controlling the function's drift inequality.
- A Brownian-bridge-style drift becomes singular near the terminal time, forcing the process toward its prescribed endpoint.
- Girsanov-based changes to drift and initialization preserve bridge conditions when the relevant squared drift adjustment is integrable.The resulting processes have the same support for measurable events under the stated assumptions.
B.1 Model Architecture for Molecule Generation.
The molecule-generation model uses an E(3)-equivariant graph neural network whose transformations respect rotations and translations. Feature rescaling improves stability performance.
- The molecule model uses an E(3)-equivariant graph neural network as its basic architecture.
- Equivariance requires predictions to transform consistently when coordinates undergo a rotation and translation.
- The network stacks equivariant graph convolutional layers operating on node coordinates and additional features.
- Scaling atom-type one-hot vectors and atom charges improves molecule stability by 47% relative to non-scaled inputs.
B.2 Model Architecture for Point Cloud Generation.
The point-cloud model follows a prior diffusion architecture with separate components for learning shape priors and denoising noisy point coordinates.
- The point-cloud model contains a flow model that learns the shape prior and an encoder-based denoiser for noisy point coordinates.
C More Visualization for Point Cloud Generation
The paper presents additional visualizations of point-cloud generation for chair and airplane classes, focusing on the best-performing Bridge-Statistic results in Figure 5.
- Figure 5 visualizes the best-performing Bridge-Statistic point-cloud generation results for chair and airplane classes.