2026 - Thesis - Neural Diffusion


Topics


Tool Kits


Attended Lectures


References



Best Modern Binding Frameworks for Large-Scale Simulation (2025 Recommendation)

  • pybind11 – Stable and widely used (PyTorch official)
  • nanobind – Faster, modern, and designed for large-scale GPU simulation
Framework Key Features Advantages Limitations Typical Use Cases
pybind11 Developed at ETH Zürich, C++11 header-only library Simple, high performance, excellent integration with PyTorch/Eigen Compile-time (static) binding; cannot dynamically update kernels ✅ General scientific research / PyTorch extensions / Geometry reconstruction
nanobind Created by Wenzel Jakob (author of pybind11) Faster (≈30% less memory), supports asynchronous CUDA streams Still under active development; not fully feature-complete ✅ HPC + next-generation ML frameworks (e.g., NVIDIA Warp, Taichi, NerfAcc)
Warp (NVIDIA) NVIDIA’s native Python→CUDA JIT framework Automatically compiles CUDA kernels, no pybind layer required Limited to NVIDIA GPUs only ✅ Large-scale physical simulations (Fluid, Diffusion, 3D Gaussian models)
cppyy CERN’s Cling-based C++ reflection binding Real-time (dynamic) binding, suitable for massive simulation codes Lower stability and slightly reduced performance ✅ Large scientific simulations (e.g., particle physics, medical imaging)
Cython Python-like syntax generating C/C++ code Fast for rapid prototyping Complicated for CUDA integration, poor scalability ⚙️ Quick prototyping for small-scale models
PyO3 + Rust CUDA Rust → Python binding framework Memory-safe, zero-copy interface New ecosystem, steep learning curve ✅ Heterogeneous parallel systems / secure simulation engines


[1/2] Physiology or Medicine

Core Principle Description Mathematical Framework
High-dimensional dynamic modeling of structural systems Whether in neuronal networks, molecular structures, or protein folding, all study the state evolution of complex network systems. Graph Theory, Dynamical Systems, Tensor / PDE Simulation
Energy optimization over probability distributions All aim to find the lowest-energy, most stable, or most probable configurations. Energy-based Models, Free Energy Minimization, Statistical Mechanics
Modeling information flow in continuous space Whether synaptic signaling, electron wave functions, or chemical bonding, all involve solving evolution equations of probability densities in continuous space. Schrödinger Equation, Fokker–Planck Equation, Diffusion Equation
Topological and graph embedding problems Both connectomes and molecular bond structures can be abstracted as graphs of nodes, edges, and weights. Graph Laplacian, Spectral Graph Theory, Graph Neural Networks (GNNs)
Approximation of many-body interactions Both electron cloud interactions and synaptic electrical signals represent nonlinear coupling in many-body systems. Mean-field Approximation, Monte Carlo Simulation, Neural PDE Solver


Chemistry Simulation

Field Core Problem Computability Tools / Methods
Chemistry Simulation Predict the most stable molecular geometry and electron distribution from a known formula (minimum energy state). Solvable but complex (requires approximation). DFT, QM/MM, Diffusion-based molecular generative models
Connectomics Reconstruct the brain’s complete neural topology and functional coupling. Extremely large-scale (≈10¹⁴ synapses). FFN, SENSE, SHAPE, GNN, Transformer
Alzheimer’s Disease Explain how structural degeneration leads to cognitive decline. Highly complex and non-deterministic (biological variability and temporal evolution). Graph Diffusion Models, Protein Misfolding Simulation, Causal Modeling


  • In the brain connectivity matrix, random matrix theory helps us identify which connectivity patterns are functional (signal) and which are just random noise (noise)
┌─────────────────────────────────┐
│  Chemistry Simulation (DFT, QM) │ ← Microscopic Level
│  → Computes atomic interactions │
└──────────────┬──────────────────┘
               ↓
┌────────────────────────────────┐
│  Connectomics                  │ ← Mesoscopic Level
│  → Maps neuron-to-neuron graph │
└──────────────┬─────────────────┘
               ↓
┌───────────────────────────────┐
│  Alzheimer’s Disease Modeling │ ← Macroscopic Level
│  → Studies functional decline │
└───────────────────────────────┘


[2/2] Connectomics - 4D Reconstruction

  • Dark Matter Detection / Chemistry Simulation

Formulations

Define the neural manifold evolution:

\[\frac{d\mathcal{M}_t}{dt} = \mathcal{F}(\mathcal{M}_t, W_t, C_t)\]

Define the topological stability functional:

\[S(\mathcal{M}_t) = \int_{\mathcal{M}_t} \kappa(x, t) \, dx\]

where ( \kappa(x, t) ) denotes local curvature or connectivity density.

Disease onset condition:

\[\exists \, t_c \; \text{s.t.} \; \frac{dS(\mathcal{M}_t)}{dt}\bigg|_{t = t_c} < -\epsilon\]

Philosophical Abstraction

Concept Intuitive Meaning
Topological invariance Structural stability of the system
Random graph m-coloring Randomization of connectivity with functional labels
Product equals zero Local structural collapse (functional failure)
Loss of constraint (“no card”) Global coupling and regulation breakdown
→ Result Disease emergence as a topological phase transition — not a linear decay


Essence

  • The onset of disease corresponds to a topological phase transition in the 4D neural manifold, where the proportion of non-functional subgraphs exceeds a critical threshold, and the global topological invariants \((\chi, \beta_k)\) undergo discontinuous change, signaling the loss of structural coherence in neural connectivity.




References