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