Geometric Deep Learning
Geometric deep learning은 geometry와 symmetry group 위에 세우는 AI layer입니다. Graph, coordinate, transformation, manifold 같은 구조를 존중하는 model을 다룹니다.
이 섹션은 순수 수학 교과서가 아니라 math foundation에서 protein, molecule, 3D structure, vision, graph-structured data용 neural architecture로 넘어가는 다리입니다.
핵심 질문은 model이 transformation group 아래에서 어떻게 행동해야 하는가입니다.
여기서 는 output representation이 어떻게 변환되어야 하는지 설명합니다.
Geometric Model Contract
Geometric deep learning note는 수학 정의, 모델 구조, 생물학적 object를 섞지 않고 아래 contract를 먼저 고정합니다.
| Part | Meaning | Typical question |
|---|---|---|
| modeled object | molecule, protein, complex, point cloud, graph, field? | |
| geometric representation | coordinates, distances, frames, graph with coordinates? | |
| transformation group | permutation, SO(3), SE(3), E(3), reflection, translation? | |
| output representation | scalar, vector, tensor, coordinate, distribution? | |
| target | property, force, pose, coordinate update, generated structure? | |
| objective | scalar loss, vector loss, coordinate loss, denoising, score, velocity? | |
| metric | RMSD, force error, property metric, validity, clash, strain? |
The core consistency condition is:
If is identity, the model is invariant. If rotates, translates, or permutes the output, the model is equivariant.
결정 패턴
Geometric model에서는 architecture를 고르기 전에 contract를 먼저 정합니다.
- Object: molecule, protein, protein-ligand complex, point cloud, graph, structure.
- Target: scalar property, ranking, distance, vector field, coordinate update, pose, generated structure.
- Group: permutation, translation, rotation, reflection, SO(3), SE(3), E(3).
- Split: 필요하면 ligand scaffold, protein family, complex pair, assay/source, time.
Group 선택은 data와 deployment setting에 모두 맞아야 합니다. Preprocessing으로 강제한 symmetry는 inference time에도 같은 정보가 있을 때만 유효합니다.
Target Type Map
Target type determines the required symmetry behavior.
| Target | Required behavior | Example |
|---|---|---|
| scalar property | invariant | energy, affinity, class score, molecular property |
| set or graph label | permutation invariant or equivariant | node classification, graph property, residue label |
| vector field | rotationally equivariant | force, displacement, velocity field |
| coordinate update | SE(3)/E(3)-equivariant | denoising coordinates, pose refinement |
| distance or contact | invariant to rigid motion | contact map, distance matrix, pair feature |
| distribution over structures | equivariant sampler or invariant density | diffusion over conformers, pose generation |
This is why a structure model cannot be judged only by architecture name. The output type, loss, and metric decide what symmetry must be preserved.
수학 배경
| 필요 | 시작점 |
|---|---|
| distance, angle, coordinate basics | Geometry |
| group actions and symmetry | Symmetry group |
| vectors, matrices, bases, eigenspaces | Linear algebra |
Symmetry
| 질문 | 시작점 |
|---|---|
| should the output stay the same after a transform? | Invariance, Invariant feature |
| should the output transform with the input? | Equivariance, Equivariant feature |
| rotations only | SO(3) |
| rotations and translations | SE(3) |
| rotations, translations, and reflections | E(3) |
Coordinate와 Distance
| 필요 | 시작점 | 일반적인 output |
|---|---|---|
| declare coordinate source and frame | Coordinate frame | valid input contract |
| use distances without orientation | Distance geometry | invariant scalar features |
| predict or refine coordinates | Coordinate update | equivariant vector/coordinate output |
Representations
| 필요 | 시작점 | 쓸 때 |
|---|---|---|
| angular basis on directions | Spherical harmonics | local orientation or angular structure matters |
| scalar/vector/tensor feature types | Irreducible representation | feature channels must transform predictably |
| equivariant message passing with typed channels | Tensor Field Network | higher-order geometric expressivity is worth the cost |
Model과 Operation
| 필요 | 시작점 |
|---|---|
| choose a geometric model family | Geometric architecture |
| graph neural network with symmetry constraints | Equivariant GNN |
Boundary With Other Sections
| If the note is about | Put it in |
|---|---|
| group action, metric space, distance, representation theory | [[math/geometry-symmetry |
| layer, message passing, coordinate update, equivariant architecture | this section |
| protein, ligand, pocket, complex geometry and leakage | [[molecular-modeling/geometry-for-structure-modeling |
| docking pose, scoring, binding interaction | [[molecular-modeling/structure-based/index |
| objective, sampler, diffusion, flow matching | [[concepts/generative-models/index |
| benchmark metric, split, validity, uncertainty | [[concepts/evaluation/index |
Claim Controls
Geometric model claims are easy to overstate. Compare methods only after checking:
| Control | Why |
|---|---|
| coordinate source | experimental, predicted, docked, generated, and minimized coordinates are not equivalent |
| graph construction | edges can leak target geometry or test-time information |
| target representation | scalar, vector, coordinate, distance, and distribution require different equivariance |
| alignment and atom mapping | RMSD and pose metrics depend on correspondence and alignment rule |
| chirality and reflection policy | E(3) and SE(3) assumptions differ for chiral chemistry |
| split unit | scaffold, protein family, complex pair, and time splits support different claims |
| compute budget | equivariant tensor features can change parameter and memory cost |
Minimal claim form:
공개 가능한 check
- coordinate source가 experimental, predicted, docked, generated, simulated, conformer-generated 중 무엇인지 적습니다.
- chirality와 stereochemistry를 보존하는지 적습니다.
- graph construction이 deployment에서 사용할 수 있는 input만 쓰는지 적습니다.
- scalar output은 invariant인지, coordinate/vector output은 equivariant인지 적습니다.
- private structure, unpublished result, host path, internal benchmark name은 쓰지 않습니다.