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Approximation of solutions of parameter-dependent problems by residual neural networks

arXiv 2607.13574 ↗First posted 15 Jul 2026math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 12 of its 42 references that are in the map. First posted on arXiv 15 Jul 2026.

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References · 12

  1. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 526
  2. 2017The Expressive Power of Neural Networks: A View from the Widthcited by 318
  3. 2019Universal Approximation with Deep Narrow Networkscited by 198
  4. 2020Deep Network Approximation for Smooth Functionscited by 163
  5. 2016Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Reviewcited by 155
  6. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 95
  7. 2020Minimum Width for Universal Approximationcited by 92
  8. 2018ResNet with one-neuron hidden layers is a Universal Approximatorcited by 78
  9. 2019Deep Learning via Dynamical Systems: An Approximation Perspectivecited by 78
  10. 2021Universal Approximation Theorems for Differentiable Geometric Deep Learningcited by 41
  11. 2018Deep, Skinny Neural Networks are not Universal Approximatorscited by 37
  12. 2020Bayesian approach to inverse scattering with topological priorscited by 2

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