# Schrodinger Neural Network and Uncertainty Quantification: Quantum Machine

arXiv: https://arxiv.org/abs/2510.23449 · HTML: https://mlsanity.com/paper/2510.23449/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 15 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Built on by (most cited first, 1 of 1)

- [Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations](https://mlsanity.com/paper/2601.02492/) (2026)

## Abstract (arXiv)

We introduce the Schrodinger Neural Network (SNN), a principled architecture for conditional density estimation and uncertainty quantification inspired by quantum mechanics. The SNN maps each input to a normalized wave function on the output domain and computes predictive probabilities via the Born rule. The SNN departs from standard parametric likelihood heads by learning complex coefficients of a spectral expansion (e . g ., Chebyshev polynomials) whose squared modulus yields the conditional density p(y|x)=left| psi x(y)right| 2 with analytic normalization. This representation confers three practical advantages: positivity and exact normalization by construction, native multimodality through interference among basis modes without explicit mixture bookkeeping, and yields closed-form (or efficiently computable) functionals-such as moments and several calibration diagnostics-as quadratic forms in coefficient space. We develop the statistical and computational foundations of the SNN, including (i) training by exact maximum-likelihood with unit-sphere coefficient parameterization, (ii) physics-inspired quadratic regularizers (kinetic and potential energies) motivated by uncertainty relations between localization and spectral complexity, (iii) scalable low-rank and separable extensions for multivariate outputs, (iv) operator-based extensions that represent observables, constraints, and weak labels as self-adjoint matrices acting on the amplitude space, and (v) a comprehensive framework for evaluating multimodal predictions. The SNN provides a coherent, tractable framework to elevate probabilistic prediction from point estimates to physically inspired amplitude-based distributions.