# Equivariant Energy-Guided SDE for Inverse Molecular Design

arXiv: https://arxiv.org/abs/2209.15408 · HTML: https://mlsanity.com/paper/2209.15408/

## Key facts

- Cited by 73 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 55 references that are in the map.
- First posted on arXiv 30 Sep 2022.

## Builds on (most cited first, 30 of 42)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders](https://mlsanity.com/paper/1802.03480/) (2018)
- [Graph Convolutional Policy Network for Goal-Directed Molecular Graph Generation](https://mlsanity.com/paper/1806.02473/) (2018)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [Grammar Variational Autoencoder](https://mlsanity.com/paper/1703.01925/) (2017)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Optimization of Molecules via Deep Reinforcement Learning](https://mlsanity.com/paper/1810.08678/) (2018)
- [Constrained Graph Variational Autoencoders for Molecule Design](https://mlsanity.com/paper/1805.09076/) (2018)
- [MoFlow: An Invertible Flow Model for Generating Molecular Graphs](https://mlsanity.com/paper/2006.10137/) (2020)
- [GraphDF: A Discrete Flow Model for Molecular Graph Generation](https://mlsanity.com/paper/2102.01189/) (2021)
- [Symmetry-adapted generation of 3d point sets for the targeted discovery of molecules](https://mlsanity.com/paper/1906.00957/) (2019)

## Built on by (most cited first, 30 of 73)

- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [One Transformer Fits All Distributions in Multi-Modal Diffusion at Scale](https://mlsanity.com/paper/2303.06555/) (2023)
- [Contrastive Energy Prediction for Exact Energy-Guided Diffusion Sampling in Offline Reinforcement Learning](https://mlsanity.com/paper/2304.12824/) (2023)
- [Crystal Structure Prediction by Joint Equivariant Diffusion](https://mlsanity.com/paper/2309.04475/) (2023)
- [TFG: Unified Training-Free Guidance for Diffusion Models](https://mlsanity.com/paper/2409.15761/) (2024)
- [MiDi: Mixed Graph and 3D Denoising Diffusion for Molecule Generation](https://mlsanity.com/paper/2302.09048/) (2023)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Generative Diffusion Models on Graphs: Methods and Applications](https://mlsanity.com/paper/2302.02591/) (2023)
- [Discrete-state Continuous-time Diffusion for Graph Generation](https://mlsanity.com/paper/2405.11416/) (2024)
- [The Blessing of Randomness: SDE Beats ODE in General Diffusion-based Image Editing](https://mlsanity.com/paper/2311.01410/) (2023)
- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [Graph Diffusion Transformers for Multi-Conditional Molecular Generation](https://mlsanity.com/paper/2401.13858/) (2024)
- [Diffusion Models and Semi-Supervised Learners Benefit Mutually with Few Labels](https://mlsanity.com/paper/2302.10586/) (2023)
- [A Survey on Graph Diffusion Models: Generative AI in Science for Molecule, Protein and Material](https://mlsanity.com/paper/2304.01565/) (2023)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Learning Joint 2D & 3D Diffusion Models for Complete Molecule Generation](https://mlsanity.com/paper/2305.12347/) (2023)
- [Diffusion Models in textitDe Novo Drug Design](https://mlsanity.com/paper/2406.08511/) (2024)
- [UniGEM: A Unified Approach to Generation and Property Prediction for Molecules](https://mlsanity.com/paper/2410.10516/) (2024)
- [Spiking Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.17046/) (2023)
- [DiffPhyCon: A Generative Approach to Control Complex Physical Systems](https://mlsanity.com/paper/2407.06494/) (2024)
- [NExT-Mol: 3D Diffusion Meets 1D Language Modeling for 3D Molecule Generation](https://mlsanity.com/paper/2502.12638/) (2025)
- [A Periodic Bayesian Flow for Material Generation](https://mlsanity.com/paper/2502.02016/) (2025)
- [A Closer Look at Parameter-Efficient Tuning in Diffusion Models](https://mlsanity.com/paper/2303.18181/) (2023)
- [SDDM: Score-Decomposed Diffusion Models on Manifolds for Unpaired Image-to-Image Translation](https://mlsanity.com/paper/2308.02154/) (2023)
- [Equivariant Neural Diffusion for Molecule Generation](https://mlsanity.com/paper/2506.10532/) (2025)
- [Efficient Planning with Latent Diffusion](https://mlsanity.com/paper/2310.00311/) (2023)
- [TAGMol: Target-Aware Gradient-guided Molecule Generation](https://mlsanity.com/paper/2406.01650/) (2024)
- [Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equation](https://mlsanity.com/paper/2402.18839/) (2024)
- [Geometry Informed Tokenization of Molecules for Language Model Generation](https://mlsanity.com/paper/2408.10120/) (2024)
- [Adaptive Diffusion Guidance via Stochastic Optimal Control](https://mlsanity.com/paper/2505.19367/) (2025)

## Abstract (arXiv)

Inverse molecular design is critical in material science and drug discovery, where the generated molecules should satisfy certain desirable properties. In this paper, we propose equivariant energy-guided stochastic differential equations (EEGSDE), a flexible framework for controllable 3D molecule generation under the guidance of an energy function in diffusion models. Formally, we show that EEGSDE naturally exploits the geometric symmetry in 3D molecular conformation, as long as the energy function is invariant to orthogonal transformations. Empirically, under the guidance of designed energy functions, EEGSDE significantly improves the baseline on QM9, in inverse molecular design targeted to quantum properties and molecular structures. Furthermore, EEGSDE is able to generate molecules with multiple target properties by combining the corresponding energy functions linearly.