2.1 KiB
Next, I want to expand on the second catogory, II. Architecture-Embedded Physics: Hard Constraints and Geometric Deep Learning. I want to examine each of the referenced papers and perform further subcategorization.
- DeepH-E3 (2023) 16 | Equivariant DFT Hamiltonian
- NequIP / MACE (2021/2024) 15 | E(3)-Equivariant Potentials
- QHNet (2023) 19 | Efficient SE(3)-Equivariance
- Neural Hamiltonian Diffusion (2025) 20 | Manifold Hamiltonian Learning
- Timrov et al. (2025) 21 | Hubbard Parameter ENN
- SpinGNN (2025) 17 | Heisenberg/Spin-Lattice GNN
- ACE Framework (2024) 15 | Atomic Cluster Expansion
- AI2DFT (2024) 22 | Differential DFT Neural Code
- Deep Potentials (2021) 17 | Density-based Descriptors
- Heisenberg Edge GNN 17 | Equivariant Message Passing
- Atomic-site ENN (2024) 15 | Lattice Symmetry-Aware
- Group-Equivariant Survey 18 | Group Representation Theory
For each of these papers, identify the core architectural insight. Try to find common subcategories, and for each subcategory, express it in terms of our unified formulation.
The driving formulation for all these models is that there is some general time-dependent nonlinear PDE of the form:
d_t u(x, t) + N_u(x, t) = 0
where N denotes a possibly nonlinear spatial differential operator, and u is the unknown physical field. The neural network u_theta approximates u, and from this we can solve inverse problems or forward inference.
PINNs define a physics residual by substituting the neural approximation u_theta into the governing equation:
r_theta(x, t) = d_t u_theta(x, t) + N_u_theta(x, t)
The governing equation is satisfied at a point (x, t) when r_theta(x, t) = 0,
so the physics loss penalizes based on this residual at various observations
(x_j, t_j).
That is, we optimize for the model parameters theta by
argmin_theta | N_theta(x, t) - u_theta(x, t) |^2_2
In architecture embedded physics, the model approximation u_theta
incorporates some physics-based differential possibly nonlinear operator
partial which informs the result.
argmin_theta | N_theta(partial, x, t) - u_theta(partial, x, t) |^2_2