Introduction
Generative modeling of 3D molecular structures, if deployed successfully, can help on many problems in material and life sciences.
Recently, state-of-the-art image and text generative models were adapted to 3D molecule generation, achieving some degree of success [ 2 , 3 ] .
However, unlike other domains where the data modality is defined by the representation itself (e.g., a digital image is a tensor of pixels), there are multiple ways to represent a molecule. Therefore, an important problem to consider when modeling 3D molecules is: what constitutes a good representation for molecules?

Recent methods for 3D molecule generation usually represent molecules as point clouds of atoms [ 4 ] or discrete grids of atomic densities [ 5 ] , which we will refer to as voxel grids.
Point clouds are processed by graph neural networks (GNNs), usually based on equivariant architectures [ 6 , 7 ] .
GNNs are known to be less expressive than other architectures due to the message passing formalism [ 8 , 9 , 10 ] and often scale quadratically with the number of atoms.
On the other hand, voxel grids are compatible with more expressive models (e.g., convnets and transformers) but computation and memory scales cubically with the volume occupied by the molecules. These limitations in expressivity and scalability hinder the scope of application of these models.

In this work, we propose a new representation for molecules that overcomes those limitations.
Inspired by the 3D computer vision community [ 11 ] , we represent molecules as fields encoding atomic occupancy , i.e., continuous functions that map 3D coordinates to atomic densities.
Arguably this representation is more natural for molecules than for visual data: while visual data is obtained via discrete measurements, molecular fields are continuous by nature.
We handle these fields as such, by parameterizing the molecular occupancy field with a neural network, shared among all molecules, and modulation codes, specific to each molecule.
Fields that are parametrized by neural networks are referred to as neural fields, implicit neural representations (INR) or coordinate-based neural networks.
The former models common molecular structures (e.g., bonds, angles, valencies, symmetries) while the later encodes variations that make each molecule unique.
Given a modulation code, we decode the molecular field by predicting the occupancy of each atom at given 3D coordinates (see Figure ˜ 1 (a)).
This decodes the molecules into explicit representations (such as discrete grids at arbitrary resolution or a .sdf format file), useful for downstream tasks.

We perform generative modeling in the continuous function space simply by sampling new modulation codes.
Our proposed approach, FuncMol , leverages a modulation code denoiser to sample molecules following the (score-based) walk-jump sampling (WJS) approach [ 1 ] .
WJS enjoys many properties such as fast-mixing, simplicity for training and fast sampling speed.
Sampling is composed of three steps: (i) (walk) sample noisy modulation codes with a Langevin Markov chain Monte Carlo (MCMC), (ii) (jump) estimate the “clean” modulation codes, and (iii) (decode) convert the estimated codes into a molecule. Figure ˜ 1 (b) illustrates a WJS chain, with samples generated by our model trained on a macrocyclic peptides dataset [ 12 ] .

The neural molecular field representation has many advantages over prior representations:
(i) it represents complex high-dimensional data in a relatively low-dimensional compact space,
(ii) it is scalable (w.r.t. the number of points, size of molecules and resolution) and has low memory footprint,
(iii) it does not make any assumptions on molecular structure or geometry,
(iv) it can represent molecular structures at arbitrary resolutions and for a free-form discretization,
(v) it is compatible with expressive machine learning architectures, and
(vi) it is domain-agnostic and can be used for a variety of molecular design problems that can be expressed over fields, e.g., atomic densities, surfaces, pharmacophores, molecular orbitals, electron densities etc.

In summary, our contributions are as follows.
We introduce a new way to represent molecular structures with neural fields. These representations are low-dimensional, compact, scalable and do not make any assumptions on the molecular structure.
We then propose FuncMol, a score-based model for 3D molecule generation that leverages these representations. We show that FuncMol performs competitively against representative baselines on the drug-like molecules dataset GEOM-drugs [ 13 ] , based on a wide set of standard and new metrics that we introduce to better measure the generation quality.
These results were achieved with one order magnitude faster sampling time. 2 2 2 Sampling time includes the “decoding” step to convert the generated code into a molecule. Finally, we illustrate FuncMol’s ability to scale to larger 3D molecules by training it on CREMP [ 12 ] , a recent macro-cyclic peptide dataset, to which our baselines are currently unable to scale.