# Schrodinger AI: A Unified Spectral-Dynamical Framework for Classification, Reasoning, and Operator-Based Generalization

arXiv: https://arxiv.org/abs/2512.22774 · HTML: https://mlsanity.com/paper/2512.22774/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 28 references that are in the map.
- First posted on arXiv 28 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Learning to Represent Programs with Graphs](https://mlsanity.com/paper/1711.00740/) (2017)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Measuring abstract reasoning in neural networks](https://mlsanity.com/paper/1807.04225/) (2018)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Neural Programmer-Interpreters](https://mlsanity.com/paper/1511.06279/) (2015)
- [Operator-Based Machine Intelligence: A Hilbert Space Framework for Spectral Learning and Symbolic Reasoning](https://mlsanity.com/paper/2507.21189/) (2025)

## Abstract (arXiv)

We introduce textbfSchr"odinger AI, a unified machine learning framework inspired by quantum mechanics. The system is defined by three tightly coupled components: (1) a time-independent wave-energy solver that treats perception and classification as spectral decomposition under a learned Hamiltonian; (2) a time-dependent dynamical solver governing the evolution of semantic wavefunctions over time, enabling context-aware decision revision, re-routing, and reasoning under environmental changes; and (3) a low-rank operator calculus that learns symbolic transformations such as modular arithmetic through learned quantum-like transition operators. Together, these components form a coherent physics-driven alternative to conventional cross-entropy training and transformer attention, providing robust generalization, interpretable semantics, and emergent topology. Empirically, Schr"odinger AI demonstrates: (a) emergent semantic manifolds that reflect human-conceived class relations without explicit supervision; (b) dynamic reasoning that adapts to changing environments, including maze navigation with real-time potential-field perturbations; and (c) exact operator generalization on modular arithmetic tasks, where the system learns group actions and composes them across sequences far beyond training length. These results suggest a new foundational direction for machine learning, where learning is cast as discovering and navigating an underlying semantic energy landscape.