Spring 2026

prompt-02-detail-research.md

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+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