Sum-of-Gaussians Tensor Neural Network (SOG-TNN)

• The paper introduces SOG-TNN, a variational ansatz that integrates tensor-product neural bases with sum-of-Gaussians kernel approximations to efficiently solve many-electron Schrödinger equations.
• Methodologically, it employs one-dimensional feedforward subnetworks and model reduction techniques like weighted balanced truncation and SVD to reduce high-dimensional integrals.
• Empirical results demonstrate spectral-like convergence and significant computational savings compared to sparse-grid and spherical-harmonic methods for systems such as H, He, Li, and Be.

Searching arXiv for the specified SOG-TNN papers and closely related antecedents. The Sum-of-Gaussians Tensor Neural Network (SOG-TNN) is a variational neural-network ansatz for Schrödinger equations that combines a low-rank tensor neural network representation of the wave function with a sum-of-Gaussians approximation of Coulomb-type interaction kernels. In the form presented for the many-electron problem, SOG-TNN couples learned one-dimensional feedforward subnetworks, tensorized basis construction, kernel separabilization, and exact fermionic antisymmetry via a Slater determinant ansatz (Wu et al., 25 Mar 2026). A broader formulation for the high-dimensional Schrödinger equation uses the same central idea—tensor-product neural bases together with separable Gaussian decompositions of Coulomb interactions—to reduce high-dimensional integration to combinations of low-dimensional integrals, supplemented by range splitting and offline model reduction (Zhou et al., 14 Aug 2025).

1. Definition and problem setting

SOG-TNN is introduced for the many-electron Schrödinger equation in settings where the main computational difficulties are high dimensionality, strong correlation, and the nonseparable character of Coulomb interactions. In the one-dimensional soft-Coulomb formulation, the target equation is

$\mathcal{H}\Psi = E\Psi,$

with Hamiltonian

$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$

where the interaction kernel is

$v(u)=\frac{1}{\sqrt{1+u^2}}.$

The wave function is required to lie in the antisymmetric fermionic space $V_{\mathcal A}$, satisfying

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$

(Wu et al., 25 Mar 2026).

In the higher-dimensional atomic setting, the corresponding Hamiltonian is

$$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$

and the method is framed through the Rayleigh principle

$$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$

(Zhou et al., 14 Aug 2025).

The defining feature of SOG-TNN is the joint use of tensor-product neural bases and Gaussian kernel decompositions. This allows the wave function to be represented in low rank while transforming Coulomb terms into separable expressions. In the one-dimensional many-electron setting, the method is explicitly described as combining a tensor neural network representation, a sum-of-Gaussians (SOG) approximation of the Coulomb kernel, and a Slater determinant ansatz to enforce antisymmetry exactly (Wu et al., 25 Mar 2026).

2. Tensor neural representation of the wave function

In the many-electron soft-Coulomb construction, the wave function is approximated as

$$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$

where $P$ is the basis size, $\alpha_p$ are linear coefficients, $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$0 are trainable parameters, $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$1 is the $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$2-th tensorized basis, and $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$3 is the antisymmetrization operator implemented by a Slater determinant (Wu et al., 25 Mar 2026).

Each one-electron orbital $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$4 is generated by a one-dimensional feedforward neural network,

$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$5

with layer map

$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$6

Accordingly, the basis functions are not fixed orbitals or polynomials; they are learned nonlinear one-dimensional neural-network features combined tensorially (Wu et al., 25 Mar 2026).

The broader high-dimensional SOG-TNN formulation adopts an analogous low-rank tensor-product expansion. For $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$7,

$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$8

where each coordinate is handled by its own one-dimensional feedforward subnetwork and the basis functions are tensor products

$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$9

This structure permits reductions such as

$v(u)=\frac{1}{\sqrt{1+u^2}}.$0

so high-dimensional integrals are replaced by products of one-dimensional integrals (Zhou et al., 14 Aug 2025).

The high-dimensional paper also states a theoretical approximation estimate:

$v(u)=\frac{1}{\sqrt{1+u^2}}.$1

with a dimension-dependent constant $v(u)=\frac{1}{\sqrt{1+u^2}}.$2 that decays like $v(u)=\frac{1}{\sqrt{1+u^2}}.$3. This is presented as a formal justification that low-rank tensor structure can mitigate the curse of dimensionality (Zhou et al., 14 Aug 2025).

3. Sum-of-Gaussians decomposition and model reduction

The kernel treatment is the central computational mechanism in SOG-TNN. In the one-dimensional soft-Coulomb setting, the interaction kernel is approximated by a bilateral Gaussian series,

$v(u)=\frac{1}{\sqrt{1+u^2}}.$4

with

$v(u)=\frac{1}{\sqrt{1+u^2}}.$5

for tunable parameters $v(u)=\frac{1}{\sqrt{1+u^2}}.$6 and $v(u)=\frac{1}{\sqrt{1+u^2}}.$7. The approximation error is stated to behave like

$v(u)=\frac{1}{\sqrt{1+u^2}}.$8

(Wu et al., 25 Mar 2026).

A first reduction stage is weighted balanced truncation (WBT), which compresses the large Gaussian sum:

$v(u)=\frac{1}{\sqrt{1+u^2}}.$9

with $V_{\mathcal A}$0. A second reduction stage applies a separable Chebyshev expansion to each Gaussian kernel factor,

$V_{\mathcal A}$1

followed by an SVD compression of the coefficient matrix $V_{\mathcal A}$2, so that the retained rank satisfies $V_{\mathcal A}$3 (Wu et al., 25 Mar 2026).

For $V_{\mathcal A}$4, $V_{\mathcal A}$5, and $V_{\mathcal A}$6, the reported reduction is from about 220 Gaussians in the raw bilinear-series approximation to only 26–28 Gaussians after WBT, and SVD further reduces the Chebyshev rank by roughly two-thirds. The specific values reported are as follows (Wu et al., 25 Mar 2026).

$V_{\mathcal A}$7	$V_{\mathcal A}$8	$V_{\mathcal A}$9
13	26	229
15	27	270
20	28	361

The high-dimensional SOG-TNN paper uses the classical Gaussian integral identity

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$0

to derive a bilateral sum-of-Gaussians approximation of the Coulomb kernel,

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$1

With

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$2

the Coulomb term becomes

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$3

which is fully separable in Cartesian coordinates (Zhou et al., 14 Aug 2025).

That paper supplements the decomposition by a range-splitting scheme dividing Gaussian indices into short-, mid-, and long-range sets. Short-range terms use an asymptotic expansion, long-range terms use low-rank Chebyshev expansion, and mid-range terms use SVD-based model reduction. This is described as the principal mechanism for reducing the number of expensive two-dimensional integrals in electron-electron interactions (Zhou et al., 14 Aug 2025).

4. Antisymmetry and the Slater determinant ansatz

For fermionic systems, SOG-TNN incorporates antisymmetry structurally rather than through penalties in the one-dimensional soft-Coulomb formulation. The antisymmetrization operator for spin-$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$4 electrons is

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$5

This is explicitly the determinant structure of a Slater determinant (Wu et al., 25 Mar 2026).

Because exchanging any two same-spin electrons changes the determinant sign, the wave function automatically satisfies

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$6

The spin sectors are handled separately according to

$T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$7

and the full wave function is formed from the product of spin-resolved antisymmetrized factors (Wu et al., 25 Mar 2026).

This strict enforcement of antisymmetry is a defining distinction of the many-electron soft-Coulomb SOG-TNN formulation. By contrast, the earlier high-dimensional SOG-TNN paper states that antisymmetry constraints are enforced for same-spin electrons via exchange operators $T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$8, and its algorithmic outline includes a loss with Pauli penalties (Zhou et al., 14 Aug 2025). This suggests that the later formulation with a Slater determinant ansatz shifts antisymmetry from a training constraint to an exact architectural property.

5. Variational workflow, complexity, and deterministic energy evaluation

In the one-dimensional soft-Coulomb application, the workflow is stated as follows: truncate the spatial domain to $T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N$9, represent the wave function by SOG-TNN with antisymmetric Slater determinants, use SOG + WBT + SVD to convert electron-electron interactions into efficiently computable one-dimensional integrals, evaluate the energy functional deterministically,

$$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$0

and optimize with a hybrid strategy consisting of RAdam with cosine annealing and warm restarts, then another cosine annealing phase, then L-BFGS fine-tuning (Wu et al., 25 Mar 2026).

After factorization, the Hamiltonian assembly cost is given as

$$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$1

where $$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$2 is the cost of the one-dimensional numerical quadrature. The substantive point is that the original high-dimensional electron-electron interaction is converted into a manageable set of one-dimensional integrals (Wu et al., 25 Mar 2026).

The high-dimensional SOG-TNN paper provides a more detailed algorithmic decomposition. Its per-step costs include

• mass matrix: $$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$3,
• kinetic term: $$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$4,
• ion-electron term:

$$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$5

• electron-electron term:

$$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$6

• eigenvalue solve: $$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$7,

with total per-step complexity

$$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$8

and memory

$$\hat H=-\frac12\sum_{i=1}^N \Delta_i +\sum_{1\le i<j\le N}\frac1{|\mathbf r_i-\mathbf r_j|} -\sum_{i=1}^N\sum_{k=1}^M\frac{Q_k}{|\mathbf r_i-\mathbf R_k|} +\sum_{1\le k<l\le M}\frac{Q_kQ_l}{|\mathbf R_k-\mathbf R_l|},$$9

These expressions formalize the claim that the method is low-memory relative to full high-dimensional discretizations or dense multidimensional interaction tables (Zhou et al., 14 Aug 2025).

6. Convergence behavior and numerical performance

The later many-electron paper reports spectral-like convergence of the relative ground-state energy error

$$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$0

with respect to basis size $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$1, using the mixed algebraic-exponential model

$$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$2

For atoms H through O, the reported fits satisfy $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$3, with values including $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$4 for H, $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$5 for He, $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$6 for Li, $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$7 for Be, $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$8 for B, $$E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}$$9 for C, $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$0 for N, and $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$1 for O (Wu et al., 25 Mar 2026).

The same paper states that chemical accuracy is reached for all tested atoms with only $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$2, and that with $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$3 the errors are below $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$4 for H, He, Li, Be, B and below $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$5 for C, N, O. Reported high-accuracy energies were obtained with compact basis sizes up to oxygen, including H with $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$6, He with $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$7, Li with $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$8, Be with $$\Psi(\bm r;\mathbf{\Theta}) = \sum_{p=1}^P \alpha_p \, \mathcal{A}\!\left(\bm{\Phi}_p^\uparrow(\bm r^\uparrow;\mathbf{\Theta}^{\uparrow})\right) \cdot \mathcal{A}\!\left(\bm{\Phi}_p^\downarrow(\bm r^\downarrow;\mathbf{\Theta}^{\downarrow})\right),$$9, B with $P$0, C with $P$1, N with $P$2, and O with $P$3 (Wu et al., 25 Mar 2026).

The comparison with sparse-grid CI (SG-CI) is reported in similarly concrete terms. For Li, SOG-TNN reaches $P$4 accuracy with $P$5, while SG-CI needs about $P$6 bases. For C, SOG-TNN gets $P$7 accuracy with $P$8, whereas SG-CI remains around $P$9 even with tens of thousands of bases (Wu et al., 25 Mar 2026).

The earlier high-dimensional SOG-TNN paper reports numerical experiments on H, He, Li, and Be. It states that the method uses two hidden layers in each 1D subnet and sine activation, with Adam and a piecewise learning-rate decay introduced when needed to avoid training rebounds. For helium, it reports computed ground-state energy

$\alpha_p$0

against the reference

$\alpha_p$1

with relative accuracy about $\alpha_p$2. For lithium, it reports

$\alpha_p$3

against the reference

$\alpha_p$4

again at about $\alpha_p$5 accuracy. For beryllium, it reports

$\alpha_p$6

and about $\alpha_p$7 accuracy (Zhou et al., 14 Aug 2025).

That paper also compares SOG-TNN against spherical-harmonic-expansion TNN (SHE-TNN). Representative values are reported as follows (Zhou et al., 14 Aug 2025).

System	SHE-TNN error	SOG-TNN error
He	$\alpha_p$8	$\alpha_p$9
Li	$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$00	$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$01
Be	$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$02	$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$03

The same comparison reports memory and time-per-step reductions: for He, $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$04 MiB versus $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$05 MiB and $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$06 ms versus $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$07 ms; for Li, $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$08 MiB versus $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$09 MiB and $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$10 ms versus $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$11 ms; for Be, $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$12 MiB versus $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$13 MiB and $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$14 ms versus $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$15 ms (Zhou et al., 14 Aug 2025).

7. Related formulations and interpretive context

SOG-TNN belongs to a broader family of tensorized neural constructions in which sums over local components are organized by low-rank tensor structure. A relevant antecedent is Tensorial Mixture Models (TMMs), which are not named SOG-TNN but are described as generative models with mixtures of local components and tensor-factorized global mixing weights, implemented by a Convolutional Arithmetic Circuit (ConvAC) (Sharir et al., 2016).

In TMMs, the input is decomposed into local parts $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$16, each governed by a hidden assignment $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$17, and the joint density is

$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$18

The prior $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$19 is interpreted as an order-$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$20 tensor $$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$21, and the paper connects its parameterization to classical tensor decompositions: CP decomposition corresponds to shallow ConvAC / shallow TMM, while Hierarchical Tucker corresponds to deep ConvAC / deep TMM (Sharir et al., 2016).

The local component family in TMMs is frequently Gaussian, and in sparse special cases the model reduces to a standard diagonal Gaussian mixture model,

$$\mathcal{H} := \sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{r_i}^2 -\sum_{J=1}^{M} z_Jv(|r_i-R_J|) + \sum_{j=i+1}^Nv(r_{ij})\right) + \sum_{I<J} z_Iz_Jv(R_{IJ}),$$22

This supports the characterization of TMMs as a strong antecedent for later architectures combining Gaussian components, tensor factorization, and neural implementation (Sharir et al., 2016).

The relationship is not identity. In TMMs, the Gaussian components are local factors in a generative density model, whereas in SOG-TNN the Gaussian structure is used primarily as a separable approximation of interaction kernels within a variational solver for Schrödinger equations. A plausible implication is that both lines of work share a tensorized “sum-over-components” logic, but they deploy it toward different computational ends: tractable marginalization and generative inference in TMMs, versus deterministic energy evaluation and low-rank many-body approximation in SOG-TNN (Sharir et al., 2016).

Taken together, the SOG-TNN literature presents a progression from a general high-dimensional tensor-neural solver with SOG kernel decomposition and range splitting (Zhou et al., 14 Aug 2025) to an improved many-electron architecture with model reduction techniques, exact Slater-determinant antisymmetry, and observed mixed algebraic-exponential spectral convergence for one-dimensional soft-Coulomb systems (Wu et al., 25 Mar 2026).