# Man, Machine, and Mathematics

arXiv: https://arxiv.org/abs/2604.27052 · HTML: https://mlsanity.com/paper/2604.27052/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 48 of its 102 references that are in the map.
- First posted on arXiv 29 Apr 2026.

## Builds on (most cited first, 30 of 48)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Pruning neural networks without any data by iteratively conserving synaptic flow](https://mlsanity.com/paper/2006.05467/) (2020)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel](https://mlsanity.com/paper/2010.15110/) (2020)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Optimizing Neural Networks via Koopman Operator Theory](https://mlsanity.com/paper/2006.02361/) (2020)
- [Analyzing Finite Neural Networks: Can We Trust Neural Tangent Kernel Theory?](https://mlsanity.com/paper/2012.04477/) (2020)
- [On the Existence of Universal Lottery Tickets](https://mlsanity.com/paper/2111.11146/) (2021)

## Abstract (arXiv)

Nonlinear models and optimization methods have successfully tackled a rapidly growing set of problems in recent years. Indeed, a relatively small toolbox of such models and methods can provide sufficient performance across a large landscape of tasks: deep learning alone has made significant recent contributions in scientific modelling, natural language processing, visual analysis, etc. A similar relationship exists between physical theories and phenomena, where many applications and observations emerge neatly from remarkably minimal foundations. It is natural to wonder if sparse unified frameworks could be built to steer discussion and discovery in the fields concerned with learning, optimization, and modelling. In this work, we posit and examine a possible outline for such a unified theory, interpreting the notion of ''learning'' in a broad sense. In particular, we pursue our goals by viewing learning as an inter-connected process on multiple levels: problem setup, choosing methods, and the analysis of their interplay via imposed optimisation dynamics. We begin by proposing a precise yet versatile definition for ''solvable'' problems. We then define the ''parametrised methods'' by which their solution(s) may be ''learned''. Our goal is to sketch a ''universal convergence theorem'', specifying how and when solvable problems become amenable to the methods chosen for them. We find these constructions reduce the study of learning down to remarkably few ideas and tools - many of which are simply adapted from existing ones in dynamical systems theory, geometry, and fundamental physics.