Neural Fields / Implicit Neural Representations Tutorial Master List

A reconciled curriculum for neural fields — continuous-coordinate function approximators (1D signals, 2D images, 3D scenes, time-varying fields) parameterised by neural networks. Covers MLP fundamentals, positional / Fourier encodings, SIREN-family architectures, multiplicative filter networks, conditioning, multi-resolution / hashgrid encodings, NeRF and successors, continuous-depth models, and applied case studies.

Companion lists:

Bayesian extensions (probabilistic SIREN, Bayesian NeRF, BNF) → ../bayesian_nns/TUTORIAL_MASTER_LIST.md Part G
Pure-GP machinery (RFF, spectral kernels, harmonic features) → ../gaussian_processes/TUTORIAL_MASTER_LIST.md
Normalizing flows (continuous-depth & invertible) → ../gaussianization/TUTORIAL_MASTER_LIST.md

Cross-listed items (RFF-as-PE, Slepian, deep RFF, continuous-depth flows) are flagged 🔁.

Legend — Source columns:

G = exists in gaussx (docs/notebooks/<name>)
P = exists in pyrox (docs/notebooks/<name>)
R = exists in research_notebook (projects/neural_fields/notebooks/<path>)
— = does not exist yet (gap)

Scope tag: 🧱 fundamental · 🔬 research · 🌉 bridge · 🔁 cross-listed

Refs: gh:<repo>#N = open GitHub issue (e.g., gh:pyrox#91) · dd:path = pyrox design_docs/pyrox/<path> · xref:BNN#X.Y / xref:GP#X.Y = pointer into companion list.

Curriculum at a glance¶

Part A — Foundations
- A.A — MLPs as continuous function approximators
- A.B — Positional / Fourier encodings
- A.C — Initialization & training dynamics
Part B — Architectures
- B.A — SIREN family
- B.B — Multiplicative Filter Networks
- B.C — Periodic / wavelet / Gabor INRs
Part C — Volumetric & 3D Scenes
- C.A — Vanilla NeRF
- C.B — NeRF variants (mip-NeRF, Plenoxels, Gaussian Splatting)
- C.C — Scene representations beyond MLP
Part D — Conditioning & Generalisation
- D.A — FiLM / Hyper-RFF
- D.B — Hypernetworks
- D.C — Meta-learning INRs (MAML, Reptile)
- D.D — Latent-modulated INRs
Part E — Spatial Encoding Variants
- E.A — Spherical / Slepian
- E.B — Multi-resolution hashgrid (Instant NGP)
- E.C — Tri-plane & factorised grids
Part F — Continuous-Depth Models
- F.A — Neural ODEs
- F.B — Neural CDEs / SDEs
Part G — Loss Constraints (deterministic)
- G.A — PDE residuals (PINN)
- G.B — Symmetry / equivariance / conservation
- G.C — Sparsity, smoothness, TV priors
Part H — Applied Case Studies

Part A — Foundations¶

A.A — MLPs as continuous function approximators¶

Key equations / models:

$f_\theta(x) = W_L\sigma(W_{L-1}\sigma(\cdots\sigma(W_1 x + b_1)\cdots) + b_{L-1}) + b_L$
Spectral bias (Rahaman 2019): standard ReLU MLPs preferentially fit low-frequency components
Universal approximation vs learnability — high-frequency targets need encoding tricks

#	Tutorial	Source	Scope	Refs / Notes
A.1	MLP as a continuous function approximator (1D / 2D)	—	🧱	GAP — pedagogical entry
A.2	Spectral bias of ReLU MLPs — fitting high-frequency targets fails	—	🧱	GAP — Rahaman et al. 2019

A.B — Positional / Fourier encodings¶

Key equations / models:

NeRF positional encoding (Mildenhall 2020): $\gamma(x) = [\sin(2^k\pi x), \cos(2^k\pi x)]_{k=0}^{L-1}$
Tancik (2020) Fourier feature mapping: $\gamma(x) = [\cos(2\pi Bx), \sin(2\pi Bx)]$ , $B\sim\mathcal{N}(0, \sigma^2 I)$
Equivalence: Fourier features $\leftrightarrow$ random features for a stationary kernel (Rahimi–Recht)

#	Tutorial	Source	Scope	Refs / Notes
A.3	Positional encoding (Tancik 2020 / NeRF Fourier features)	—	🧱	GAP
A.4	Gaussian-feature INRs (RFF as positional encoding)	—	🧱 🔁	GAP — bridges `xref:BNN#A.8` (RFF) ↔ this list’s B.1 (SIREN)
A.5	Frequency bandwidth & lengthscale tuning for PE	—	🧱	GAP

A.C — Initialization & training dynamics¶

#	Tutorial	Source	Scope	Refs / Notes
A.6	SIREN-style $\sqrt{6/\mathrm{fan\_in}}$ init — preserve activation distribution	—	🧱	GAP
A.7	Lazy training & feature learning in INRs	—	🌉	GAP

Part B — Architectures¶

B.A — SIREN family¶

Key equations / models:

SIREN (Sitzmann 2020): $f(x) = W_L\sin(\omega_0(W_{L-1}\sin(\cdots\omega_0(W_1 x + b_1)\cdots) + b_{L-1})) + b_L$
$\omega_0 \approx 30$ controls frequency content
Derivatives are themselves SIRENs → natural for PDE / SDF supervision

#	Tutorial	Source	Scope	Refs / Notes
B.1	SIREN — sinusoidal implicit neural representations	P `siren_inr`	🌉	move research-scale → projects/neural_fields
B.2	MultiScaleSIREN — frequency-banded sinusoidal nets	—	🌉	GAP — gh:pyrox#91
B.3	SIREN derivative supervision — fit $f$ via $\nabla f$ / $\Delta f$	—	🧱	GAP

B.B — Multiplicative Filter Networks¶

Key equations / models:

MFN (Fathony 2021): $z_l = (W_l z_{l-1} + b_l)\odot g_l(x)$ with $g_l$ a Fourier/Gabor filter on $x$
FourierNet: $g_l(x) = \sin(\omega_l x + \phi_l)$ · GaborNet: $g_l(x) = e^{-\gamma\|x-\mu_l\|^2}\sin(\omega_l(x-\mu_l))$

#	Tutorial	Source	Scope	Refs / Notes
B.4	Multiplicative Filter Networks (FourierNet, GaborNet)	—	🌉	GAP — gh:pyrox#87

B.C — Periodic / wavelet / Gabor INRs¶

#	Tutorial	Source	Scope	Refs / Notes
B.5	WIRE — Gabor wavelet INR	—	🌉	GAP — Saragadam et al. 2023
B.6	Periodic INRs on the torus / sphere	—	🧱	GAP

Part C — Volumetric & 3D Scenes¶

C.A — Vanilla NeRF¶

Key equations / models:

Volume rendering: $C(\mathbf{r}) = \int_{t_n}^{t_f} T(t)\sigma(\mathbf{r}(t))c(\mathbf{r}(t), \mathbf{d})\,dt$ , $T(t) = \exp(-\int_{t_n}^t \sigma(\mathbf{r}(s))\,ds)$
Hierarchical coarse-to-fine sampling along rays

#	Tutorial	Source	Scope	Refs / Notes
C.1	NeRF — vanilla volumetric rendering	—	🔬	GAP
C.2	Camera-pose / ray-marching mechanics	—	🔬	GAP

C.B — NeRF variants¶

#	Tutorial	Source	Scope	Refs / Notes
C.3	mip-NeRF — integrated positional encoding for anti-aliasing	—	🔬	GAP
C.4	Plenoxels — explicit voxel-grid radiance fields	—	🔬	GAP
C.5	Gaussian Splatting — anisotropic Gaussians instead of MLPs	—	🔬	GAP
C.6	TensoRF / Tri-plane — factorised volumetric grids	—	🔬	GAP

C.C — Beyond MLPs¶

#	Tutorial	Source	Scope	Refs / Notes
C.7	SDF parameterisations — signed-distance fields with INRs	—	🔬	GAP
C.8	Occupancy networks	—	🔬	GAP

Part D — Conditioning & Generalisation¶

D.A — FiLM / Hyper-RFF¶

Key equations / models:

FiLM (Perez 2018): $y = \gamma(c)\odot h + \beta(c)$ injects condition $c$ via per-channel affine
Hyper-RFF: $\omega = h(c)$ — condition predicts the RFF frequencies

#	Tutorial	Source	Scope	Refs / Notes
D.1	Conditional neural fields — FiLM, Hyper-RFF	P `conditioning`	🧱

D.B — Hypernetworks¶

#	Tutorial	Source	Scope	Refs / Notes
D.2	Hypernetworks for INRs	—	🔬	GAP — research_notebook only
D.3	Set / function-space hypernetworks (DeepSets, Set Transformer)	—	🔬	GAP

D.C — Meta-learning¶

#	Tutorial	Source	Scope	Refs / Notes
D.4	MAML / Reptile for fast INR adaptation	—	🔬	GAP
D.5	Learned initialisations for INRs (Sitzmann 2020 §5)	—	🔬	GAP

D.D — Latent-modulated INRs¶

#	Tutorial	Source	Scope	Refs / Notes
D.6	Latent-modulated INRs — $f_\theta(x; z)$ with $z$ shared per signal	—	🔬	GAP
D.7	Auto-decoder training (DeepSDF-style)	—	🔬	GAP

Part E — Spatial Encoding Variants¶

E.A — Spherical / Slepian¶

#	Tutorial	Source	Scope	Refs / Notes
E.1	Slepian positional encodings — spherical, localized	—	🌉 🔁	GAP — gh:pyrox#125; xref:GP#7.19
E.2	Spherical-harmonic INRs on $S^2$	—	🔬 🔁	GAP — xref:GP#7.21

E.B — Multi-resolution hashgrid¶

#	Tutorial	Source	Scope	Refs / Notes
E.3	Hashgrid / multi-resolution encoding (Instant NGP)	—	🔬	GAP — Müller et al. 2022
E.4	Hashgrid collisions & spatial-frequency trade-offs	—	🔬	GAP

E.C — Tri-plane & factorised grids¶

#	Tutorial	Source	Scope	Refs / Notes
E.5	Tri-plane encoding (EG3D)	—	🔬	GAP
E.6	Tensorial decompositions for spatial encodings	—	🔬	GAP

Part F — Continuous-Depth Models¶

F.A — Neural ODEs¶

Key equations / models:

Neural ODE (Chen 2018): $\tfrac{dh}{dt} = f_\theta(h, t)$ , adjoint method for $O(1)$ -memory backprop
ANODE — augmented Neural ODEs for non-homeomorphic transformations

#	Tutorial	Source	Scope	Refs / Notes
F.1	Neural ODE — basic mechanics + adjoint	—	🔬	GAP
F.2	ANODE / augmented Neural ODE	—	🔬	GAP

F.B — Neural CDEs / SDEs¶

#	Tutorial	Source	Scope	Refs / Notes
F.3	Neural CDE for irregular time series	—	🔬	GAP
F.4	Neural SDE — stochastic continuous-depth	—	🔬 🔁	GAP — bridge to gaussianization list

Part G — Loss Constraints (deterministic)¶

Bayesian / probabilistic versions of these losses live in BNN list Part A.G. This section covers the deterministic PINN / equivariance / regularisation lineage.

G.A — PDE residuals¶

#	Tutorial	Source	Scope	Refs / Notes
G.1	PINN — Burgers / heat / shallow-water	—	🔬	GAP
G.2	XPINN / domain-decomposed PINNs	—	🔬	GAP
G.3	Adaptive loss weighting (Wang 2022 NTK-based)	—	🔬	GAP

G.B — Symmetry / equivariance / conservation¶

#	Tutorial	Source	Scope	Refs / Notes
G.4	Equivariant INRs (rotation, translation)	—	🔬	GAP
G.5	Divergence-free / curl-free vector-field INRs	—	🔬	GAP
G.6	Conservation laws as soft penalties (mass, momentum, energy)	—	🔬	GAP

G.C — Sparsity, smoothness, TV priors¶

#	Tutorial	Source	Scope	Refs / Notes
G.7	Total-variation / smoothness penalties	—	🌉	GAP
G.8	Sparsity-promoting regularisation (L1, group lasso)	—	🌉	GAP
G.9	Boundary-condition / initial-condition penalties	—	🔬	GAP

Part H — Applied Case Studies (research_notebook/projects/neural_fields)¶

H.A — Signals & images¶

#	Tutorial	Source	Scope	Refs / Notes
H.1	SIREN on real images / signed distance fields	P `siren_inr` (port + extend)	🔬
H.2	Image fitting with NeRF-PE vs SIREN vs MFN — comparison	—	🔬	GAP
H.3	Audio fitting with INRs	—	🔬	GAP

H.B — 3D scenes¶

#	Tutorial	Source	Scope	Refs / Notes
H.4	NeRF on a small synthetic scene	—	🔬	GAP
H.5	Gaussian Splatting on real-world capture	—	🔬	GAP

H.C — Scientific / geospatial¶

#	Tutorial	Source	Scope	Refs / Notes
H.6	Climate / SST field reconstruction with INRs	—	🔬	GAP
H.7	Spherical INRs for global atmospheric variables	—	🔬	GAP
H.8	Spatiotemporal INRs — coordinates $(x, y, t)$	—	🔬	GAP

Cross-list summary¶

Item	NF ID	Other list	Suggested home
RFF / Gaussian features as PE	A.4	BNN A.8	pyrox (BNN canonical)
Slepian PE	E.1	GP 7.19, BNN B.9	pyrox
Spherical-harmonic INRs	E.2	GP 7.18	research_notebook
Neural SDE	F.4	gaussianization	research_notebook
Bayesian INR / NeRF	—	BNN G.1–G.3	research_notebook

Proposed final homes¶

pyrox/docs/notebooks/ → A.B (PE primitives), B.A (SIREN), B.B (MFN), D.A (conditioning), E.A (Slepian)
research_notebook/projects/neural_fields/ → all of C (NeRF & variants), D.B–D.D, E.B–E.C, F, G, H

In-scope vs aspirational¶

In scope today: B.1 (SIREN exists in pyrox), D.1 (conditioning exists in pyrox)
In scope with planned features: A.3, A.4, A.6, B.2, B.3, B.4, D.2, E.1, G.1
Aspirational: everything else — needs new infra / research / dataset work