---
title: Sum-of-Gaussians Tensor Neural Network (SOG-TNN)
url: https://www.emergentmind.com/topics/sum-of-gaussians-tensor-neural-network-sog-tnn
type: topic
---

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

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 [2603.23897]. 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 [2508.10454].

## 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
$$
T_{ij}\Psi=-\Psi,\qquad 1\le i\neq j\le N
$$
[2603.23897].

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
$$
E_0=\min_{\Psi\in V_\mathcal P}\mathscr E[\Psi],\qquad \mathscr E[\Psi]=\frac{\langle\Psi|\hat H|\Psi\rangle}{\langle\Psi|\Psi\rangle}
$$
[2508.10454].

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 [2603.23897].

## 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, \(\mathbf{\Theta}=[\mathbf{\Theta}^\uparrow,\mathbf{\Theta}^\downarrow]\) are trainable parameters, \(\bm{\Phi}_p=(\phi_{p,1},\dots,\phi_{p,N})^\top\) is the \(p\)-th tensorized basis, and \(\mathcal A(\cdot)\) is the antisymmetrization operator implemented by a Slater determinant [2603.23897].

Each one-electron orbital \(\phi_{p,i}(x)\) is generated by a one-dimensional feedforward neural network,
$$
\bm{\phi}_{i}(x;\bm{\theta}_i)= \bigl[\bm{F}_{i}^{(l_{\max})}\circ \bm{F}_{i}^{(l_{\max}-1)}\circ \cdots \circ \bm{F}_{i}^{(1)}\bigr](x),
$$
with layer map
$$
\bm{F}_{i}^{(l)}(\bm X)=\bm{\eth}^{(l)}\!\bigl(\bm W_i^{(l)}\bm X+\bm b_i^{(l)}\bigr).
$$
Accordingly, the basis functions are not fixed orbitals or polynomials; they are learned nonlinear one-dimensional neural-network features combined tensorially [2603.23897].

The broader high-dimensional SOG-TNN formulation adopts an analogous low-rank tensor-product expansion. For \(d=3N\),
$$
\Psi(\mathbf x)\approx \Psi_{\mathcal T^p}(\mathbf x;\boldsymbol\theta) =\sum_{t=1}^p \alpha_t \prod_{i=1}^d \phi_{i,t}(x_i;\boldsymbol\theta_i),
$$
where each coordinate is handled by its own one-dimensional feedforward subnetwork and the basis functions are tensor products
$$
\Phi_t(\mathbf x;\boldsymbol\theta)=\prod_{i=1}^d \phi_{i,t}(x_i;\boldsymbol\theta_i).
$$
This structure permits reductions such as
$$
\int_\Omega \Psi(\mathbf x)\,d\mathbf x \approx \sum_{t=1}^p \alpha_t \prod_{i=1}^d \int_{\Omega_i}\phi_{i,t}(x_i)\,dx_i,
$$
so high-dimensional integrals are replaced by products of one-dimensional integrals [2508.10454].

The high-dimensional paper also states a theoretical approximation estimate:
$$
\|\Psi-\Psi_{\mathcal T^p}\|_{H^m(\Omega^d)} \le C(d)\,p^{-(s-m+t)}\|\Psi\|_{H_{\mathrm{mix}^{t,s}(\Omega^d)}},
$$
with a dimension-dependent constant \(C(d)\) that decays like \(c\,d^2\,0.97515^d\). This is presented as a formal justification that low-rank tensor structure can mitigate the curse of dimensionality [2508.10454].

## 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,
$$
v(u)\approx \sum_{\ell=-\infty}^{\infty}\omega_\ell\,e^{-s_\ell u^2},
$$
with
$$
s_\ell=\frac{1}{2b^{2\ell}\sigma^2}, \qquad \omega_\ell=\frac{2\ln b\,e^{-s_\ell}}{\sqrt{2\pi}b^\ell\sigma},
$$
for tunable parameters \(b>1\) and \(\sigma>0\). The approximation error is stated to behave like
$$
\left|1-\frac{1}{v(u)}\sum_{\ell=-\infty}^{\infty}\omega_\ell e^{-s_\ell u^2}\right| \lesssim 2\sqrt{2}\exp\!\left(-\frac{\pi^2}{2\ln b}\right)
$$
[2603.23897].

A first reduction stage is **weighted balanced truncation (WBT)**, which compresses the large Gaussian sum:
$$
\left| \sum_{\ell=-L_1}^{L_2} \omega_\ell e^{-s_\ell u^2} - \sum_{\ell=1}^{L} \widetilde{\omega}_\ell e^{-\widetilde{s}_\ell u^2} \right|<\epsilon,
$$
with \(L\ll L_1+L_2\). A second reduction stage applies a separable Chebyshev expansion to each Gaussian kernel factor,
$$
e^{-\widetilde s_\ell (r-r')^2} \approx \sum_{m+n\le S} C_{m,n}^{\ell} T_m\!\left(\frac{r}{r_c}\right) T_n\!\left(\frac{r'}{r_c}\right),
$$
followed by an SVD compression of the coefficient matrix \(\mathbf C^\ell\), so that the retained rank satisfies \(K^\ell \ll S\) [2603.23897].

For \(\epsilon=10^{-12}\), \(\sigma=1\), and \(b=1.187800189600461\), 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 [2603.23897].

| \(r_c\) | \(L\) | \(K_{\max}\) |
|---|---:|---:|
| 13 | 26 | 229 |
| 15 | 27 | 270 |
| 20 | 28 | 361 |

The high-dimensional SOG-TNN paper uses the classical Gaussian integral identity
$$
\frac1r = \frac{2}{\sqrt\pi}\int_0^\infty e^{-x^2 r^2}\,dx
$$
to derive a bilateral sum-of-Gaussians approximation of the Coulomb kernel,
$$
\frac1r \approx \frac{2\ln b}{\sqrt{2\pi}\sigma} \sum_{\ell=-\infty}^{\infty} \frac{1}{b^\ell} \exp\!\left[-\frac12\left(\frac{r}{b^\ell\sigma}\right)^2\right].
$$
With
$$
s_\ell=b^\ell\sigma, \qquad w_\ell = 4\pi s_\ell^2 \ln b, \qquad G_\ell(x)=\frac{1}{\sqrt{2\pi}s_\ell}\exp\!\left(-\frac{x^2}{2s_\ell^2}\right),
$$
the Coulomb term becomes
$$
\frac{1}{|\mathbf r_i-\mathbf R_k|} \approx \sum_{\ell=-M_2}^{M_1} w_\ell\, G_\ell(x_i-X_k)\,G_\ell(y_i-Y_k)\,G_\ell(z_i-Z_k),
$$
which is fully separable in Cartesian coordinates [2508.10454].

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 [2508.10454].

## 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-\(*\) electrons is
$$
\mathcal{A}\!\left(\bm{\Phi}_p^*(\mathbf r^*;\mathbf\Theta^*)\right) := \frac{1}{N^*!} \left| \begin{array}{ccc} \phi_{p,\tau^*(1)}(r_{\tau^*(1)}) & \cdots & \phi_{p,\tau^*(1)}(r_{\tau^*(N^*)})\\
\vdots & \ddots & \vdots\\
\phi_{p,\tau^*(N^*)}(r_{\tau^*(1)}) & \cdots & \phi_{p,\tau^*(N^*)}(r_{\tau^*(N^*)})
\end{array} \right|.
$$
This is explicitly the determinant structure of a Slater determinant [2603.23897].

Because exchanging any two same-spin electrons changes the determinant sign, the wave function automatically satisfies
$$
T_{ij}\Psi=-\Psi.
$$
The spin sectors are handled separately according to
$$
N^\uparrow=\lceil N/2\rceil,\qquad N^\downarrow=N-N^\uparrow,
$$
and the full wave function is formed from the product of spin-resolved antisymmetrized factors [2603.23897].

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 \(T_{ij}\), and its algorithmic outline includes a loss with Pauli penalties [2508.10454]. 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 \([-r_c,r_c]^N\), 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,
$$
E_0=\min_{\Psi\in V_{\mathcal A}} \frac{\langle \Psi|\mathcal H|\Psi\rangle}{\langle \Psi|\Psi\rangle},
$$
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** [2603.23897].

After factorization, the Hamiltonian assembly cost is given as
$$
\mathcal O(P^2 N^3 L K_{\max} T_{\text{1D}}),
$$
where \(T_{\text{1D}}\) 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 [2603.23897].

The high-dimensional SOG-TNN paper provides a more detailed algorithmic decomposition. Its per-step costs include
- mass matrix: \(\mathcal O(3p^2NK T_{\rm eval})\),
- kinetic term: \(\mathcal O(9p^2NK T_{\rm eval})\),
- ion-electron term:
  $$
  \mathcal C_{ie}
  =
  \mathcal O\!\left(3p^2MN + 3|\mathcal M|p^2MNK T_{\rm eval}\right),
  $$
- electron-electron term:
  $$
  \mathcal C_{ee}
  =
  \mathcal O\!\left(
  3p^2N^2KT_{\rm eval}
  +
  3|\mathcal M_{\text{long}}|(S+1)p^2NKT_{\rm eval}
  +
  S^2N^2
  +
  3\sum_{\ell\in\mathcal M_{\text{mid}}}p^2N^2r_\ell K T_{\rm eval}
  \right),
  $$
- eigenvalue solve: \(\mathcal O(p^3)\),

with total per-step complexity
$$
\mathcal C_{\rm total} = \mathcal O\!\left( p^3 + S^2N^2 + (S|\mathcal M_{\rm long}| + N|\mathcal M_{\rm mid}| + M|\mathcal M|)p^2KNT_{\rm eval} \right),
$$
and memory
$$
\mathcal O(pNK+p^2+K+S^2+SK+M).
$$
These expressions formalize the claim that the method is low-memory relative to full high-dimensional discretizations or dense multidimensional interaction tables [2508.10454].

## 6. Convergence behavior and numerical performance

The later many-electron paper reports **spectral-like convergence** of the relative ground-state energy error
$$
\mathcal{E}_r=\frac{E-E_0}{|E_0|}
$$
with respect to basis size \(P\), using the mixed algebraic-exponential model
$$
\mathcal{E}_r = C\,P^{-\beta}e^{-\gamma P}.
$$
For atoms H through O, the reported fits satisfy \(R^2\ge 0.989\), with values including \(R^2=1.00\) for H, \(0.996\) for He, \(0.989\) for Li, \(0.996\) for Be, \(0.998\) for B, \(0.994\) for C, \(0.997\) for N, and \(0.999\) for O [2603.23897].

The same paper states that **chemical accuracy** is reached for all tested atoms with only \(P\le 20\), and that with \(P=60\) the errors are below \(10^{-7}\) for H, He, Li, Be, B and below \(10^{-5}\) for C, N, O. Reported high-accuracy energies were obtained with compact basis sizes up to oxygen, including H with \(P=12\), He with \(P=20\), Li with \(P=40\), Be with \(P=60\), B with \(P=64\), C with \(P=80\), N with \(P=88\), and O with \(P=84\) [2603.23897].

The comparison with sparse-grid CI (SG-CI) is reported in similarly concrete terms. For **Li**, SOG-TNN reaches \(10^{-8}\) accuracy with \(P=20\), while SG-CI needs about \(5\times10^4\) bases. For **C**, SOG-TNN gets \(10^{-7}\) accuracy with \(P=80\), whereas SG-CI remains around \(10^{-2}\) even with tens of thousands of bases [2603.23897].

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
$$
E_0=-2.9037241864374510\ \text{a.u.}
$$
against the reference
$$
-2.9037243770341144\ \text{a.u.},
$$
with relative accuracy about \(10^{-7}\). For lithium, it reports
$$
E_0=-7.4780595868441369\ \text{a.u.}
$$
against the reference
$$
-7.478060326(10)\ \text{a.u.},
$$
again at about \(10^{-7}\) accuracy. For beryllium, it reports
$$
E_0=-14.6672437863217056\ \text{a.u.}
$$
and about \(10^{-5}\) accuracy [2508.10454].

That paper also compares SOG-TNN against spherical-harmonic-expansion TNN (SHE-TNN). Representative values are reported as follows [2508.10454].

| System | SHE-TNN error | SOG-TNN error |
|---|---:|---:|
| He | \(8.38\times 10^{-8}\) | \(6.56\times 10^{-8}\) |
| Li | \(7.78\times 10^{-7}\) | \(9.80\times 10^{-8}\) |
| Be | \(3.98\times 10^{-2}\) | \(7.68\times 10^{-6}\) |

The same comparison reports memory and time-per-step reductions: for He, \(26338\) MiB versus \(3790\) MiB and \(132\) ms versus \(69\) ms; for Li, \(49332\) MiB versus \(9070\) MiB and \(285\) ms versus \(142\) ms; for Be, \(75646\) MiB versus \(8528\) MiB and \(476\) ms versus \(199\) ms [2508.10454].

## 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) [1610.04167].

In TMMs, the input is decomposed into local parts \(X=(x_1,\ldots,x_N)\), each governed by a hidden assignment \(d_i\in[M]\), and the joint density is
$$
P(X) = \sum_{d_1,\ldots,d_N=1}^M P(d_1,\ldots,d_N) \prod_{i=1}^N P(x_i \mid d_i; \theta_{d_i}).
$$
The prior \(P(d_1,\ldots,d_N)\) is interpreted as an order-\(N\) tensor \(\mathcal A_{d_1,\ldots,d_N}\), 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 [1610.04167].

The local component family in TMMs is frequently Gaussian, and in sparse special cases the model reduces to a standard diagonal Gaussian mixture model,
$$
P(X)=\sum_{k=1}^K w_k \prod_{i=1}^N \mathcal{N}(x_i;\mu_{ki},\mathrm{diag}(\sigma^2_{ki})) = \sum_{k=1}^K w_k \mathcal{N}(X;\tilde\mu_k,\mathrm{diag}(\tilde\sigma_k^2)).
$$
This supports the characterization of TMMs as a strong antecedent for later architectures combining Gaussian components, tensor factorization, and neural implementation [1610.04167].

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 [1610.04167].

Taken together, the SOG-TNN literature presents a progression from a general high-dimensional tensor-neural solver with SOG kernel decomposition and range splitting [2508.10454] 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 [2603.23897].

Source: https://www.emergentmind.com/topics/sum-of-gaussians-tensor-neural-network-sog-tnn