# Consistency of Learned Sparse Grid Quadrature Rules using NeuralODEs

arXiv: https://arxiv.org/abs/2507.01533 · HTML: https://mlsanity.com/paper/2507.01533/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 53 references that are in the map.
- First posted on arXiv 2 Jul 2025.

## Builds on (most cited first, 14 of 14)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [A Convenient Infinite Dimensional Framework for Generative Adversarial Learning](https://mlsanity.com/paper/2011.12087/) (2020)
- [LU-Net: Invertible Neural Networks Based on Matrix Factorization](https://mlsanity.com/paper/2302.10524/) (2023)
- [New advances in universal approximation with neural networks of minimal width](https://mlsanity.com/paper/2411.08735/) (2024)
- [Distribution learning via neural differential equations: minimal energy regularization and approximation theory](https://mlsanity.com/paper/2502.03795/) (2025)
- [Numerical and statistical analysis of NeuralODE with Runge-Kutta time integration](https://mlsanity.com/paper/2503.10729/) (2025)
- [Learning Brenier Potentials with Convex Generative Adversarial Neural Networks](https://mlsanity.com/paper/2504.19779/) (2025)

## Abstract (arXiv)

We prove consistency of a recently proposed scheme that evaluates expected values by composing a learned transport map with Clenshaw--Curtis sparse-grid quadrature on a tractable product source. Our analysis hinges on the structural fact that composition of a Ckmix-regular function -- which carries the fast quadrature rate m-k(log m)(d-1)(k+1) -- with a C1-diffeomorphism can only be guaranteed to be Ckmix itself, if the diffeomorphism is diagonal up to a permutation of coordinates. The fast rate is therefore available exclusively for product targets, and the analysis splits into two regimes. In the general regime of arbitrary targets, we learn the transport as the time-one flow of a ReLUk+1-neural ODE trained by maximum likelihood. The resulting flow lies in the isotropic space Ck and yields the rate m-k/d(log m)(d-1)(k/d+1), with raising the density smoothness k and the matched activation order k+1 mitigating the curse of dimensionality at the cost of harder optimization. In the diagonal regime of product targets, the Knothe--Rosenblatt map is itself diagonal and we estimate it pointwise via empirical quantile transport, a lightweight alternative that recovers the full mixed-regularity rate. In both regimes, the resulting LtI estimator is PAC (probably approximately correct) consistent. With high probability the numerical integral approximates the true value to arbitrary accuracy as both the sample size n and the quadrature budget m tend to infinity.