Neural Structure Fields with Application toarXiv:2212.13120
Crystal Structure Autoencoders
Naoya Chiba*, Yuta Suzuki**, Tatsunori Taniai*, Ryo Igarashi*, Yoshitaka Ushiku*, Kotaro Saito***, Kanta Ono*****OMRON SINIC X Corporation, **TOYOTA Motor Corporation, ***Randeft, Inc., ****Osaka University

Neural Structure Fields (NeSF): Neural Field-based crystal structure decoder
Representing crystal structures of materials to facilitate determining them via neural networks is crucial for enabling machine-learning applications involving crystalstructure estimation. Here we propose Neural Structure Fields (NeSF) as an accurate and practical approach for representing crystal structures using neural networks.
NeSF considers a crystal structure as a continuous field rather than as a discrete set of atoms. NeSF overcomes the tradeoff between spatial resolution andcomputational complexity. NeSF can represent any crystal structure.
••••

We developed neural field-based (voxel-free) crystal decoder
Crystal structure can be written by proposed vector fields representations: Structure Fields
Neural Field technique to model a structure field by a simple neural network: NeSF
We also develop an algorithm that reconstructs atom positions and species from NeSF (or Structure Fields)

Voxel-based vs Continuous Representation

Neural Structure Field (NeSF)

continuous field representation for crystal structure

Structure Field = Position Field + Species Field
NeSF: Modeling Structure Field using Neural Field technique

Our idea of representing crystal structures as continuous vector fields has
been partially and implicitly explored using voxelization in recent MIstudy [1], but without explicit consideration as discretized vector fields. The
discretization of 3D data considerably suffers from the tradeoff between
spatial resolution and computational complexity. The proposed NeSF isinspired by the concepts of Neural Field [2, 3, 4, 5]. Theoretically, the NeSF
can achieve infinitely high spatial resolution with compact (memory- andparameter-efficient) neural networks in place of costly 3D CNNs. There isessentially no tradeoff between the spatial resolution and required memory.

• Position Field: a vector pointing to the nearest atom from a query point.
• Species Field: a one-hot vector of the nearest atom species
We approximate the Structure Field by using neural networks via Neural Fieldtechnique. We also combine PointNet-based simple encoder to the NeSF decoder,which give an autoencoder for crystal structures.

Voxel-based representation• Limited resolution (practically 32x32x32 in practice [1])• Have tradeoff between resolution and computational costs (both timeand memory)• Many parameters (often causes overfittings)
Continuous (Neural Field-based) representation• Can represents infinitely high resolution• Have no tradeoff between resolution and computational costs• Less parameters
NeSF

Neural Field techniquefP

Implicitly represents “field” by using a neural network

fS

Neural Field is a technique that represents a (vector) field by using simple
neural networks [2, 3, 4, 5]. The key idea of neural field is that the field is
implicitly described point-by-point. When a query point is given, the network
returns the value of the query point.

Input structure𝒚 = 𝑓 𝒙, 𝒛
Vector value
corresponding to- latent vector 𝒛- query point 𝒙

Query point

Encoder

Reconstructed structure(Lattice decoder)

Experimental Results
We compared proposed method with the ICSG3D as a autoencoder. On theICSG3D using dataset (AB, ABX2, and ABX3 entries on the Material Project) theproposed method achieves high accuracy to reconstruct for the test entries.

Latent vector(given field-by-field)

Modeled by neural network

Reconstruction Results in Test Dataset

MethodNeSF (Proposed)

Reconstruction Algorithm

Error in Number of Atoms [%]0.53 ± 0.25

2.67 ± 0.84

Position Error (Actual) [Å]0.0308 ± 0.0112

0.0877 ± 0.0306

Method to convert a crystal structure from a NeSF representation

Position Error (Detected) [Å]0.359 ± 0.0226

0.1057 ± 0.0284

Species Error (Actual) [%]4.31 ± 0.39

64.39 ± 1.91

1. Initialize particles: Regularly spread initial query points at 3D grid
points within a bounding box (common to each dataset)
2. Move particles: Update the position of each query points using theposition field. (Iterate this process)3. Score particles: Score each query points and filter outliers by using
estimated residual distance to the atom positions.
4. Detect atoms: Apply a simple clustering algorithm to detect eachatomic position.
5. Estimate species: to estimate the atomic species by using Species
Field at each atomic position.

Species Error (Detected) [%]4.36 ± 0.39

65.05 ± 1.85

(2)

(3)

CTm
TlGd

Ground Truth

NeSF (Proposed)

ICSG3D [1]Example of Reconstruction Result in Test Dataset: mp-22556 (Tm3TlC)

(4)

(5)

Fe

Probability(1)

ICSG3D [1]SiFe
Element

Probability

IterateCu

Position field fPCu

References[1] Court, C. J., Yildirim+ (2020). "3-D Inorganic Crystal Structure Generation and Property Prediction via Representation Learning." In: J. Chem. Inf. Model. Vol. 60, pp. 4518-4535.
[2] Mescheder, L., Oechsle+ (2019). "A. Occupancy networks: Learning 3D reconstruction in function space." In: IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).
[3] Park, J. J.+ (2019). "DeepSDF: Learning continuous signed distance functions for shape representation." In: IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).
[4] Mildenhall, B.+ (2020). "NeRF: Representing scenes as neural radiance fields for view synthesis." In: European Conference on Computer Vision (ECCV).
[5] Xie, Y.+. (2022). "Neural fields in visual computing and beyond". Comput. Graph. Forum. Vol. 41, pp. 641-676.

ElementSpecies field fS

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