# Neural Operators Can Play Dynamic Stackelberg Games

arXiv: https://arxiv.org/abs/2411.09644 · HTML: https://mlsanity.com/paper/2411.09644/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 94 references that are in the map.
- First posted on arXiv 14 Nov 2024.

## Builds on (most cited first, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities](https://mlsanity.com/paper/2210.01074/) (2022)
- [HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork](https://mlsanity.com/paper/2312.15949/) (2023)
- [Calibrated Stackelberg Games: Learning Optimal Commitments Against Calibrated Agents](https://mlsanity.com/paper/2306.02704/) (2023)
- [Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer](https://mlsanity.com/paper/2201.13094/) (2022)
- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Deep learning architectures for nonlinear operator functions and nonlinear inverse problems](https://mlsanity.com/paper/1912.11090/) (2019)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems](https://mlsanity.com/paper/1903.06652/) (2019)
- [Oracles & Followers: Stackelberg Equilibria in Deep Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2210.11942/) (2022)

## Built on by (most cited first, 5 of 5)

- [Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs](https://mlsanity.com/paper/2508.14995/) (2025)
- [One model to solve them all: 2BSDE families via neural operators](https://mlsanity.com/paper/2511.01125/) (2025)
- [Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operators](https://mlsanity.com/paper/2510.20017/) (2025)
- [Deep Neural Operator Learning for Probabilistic Models](https://mlsanity.com/paper/2511.07235/) (2025)
- [NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes](https://mlsanity.com/paper/2607.14361/) (2026)

## Abstract (arXiv)

Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the textitfollower's best-response operator can be approximately implemented by an textitattention-based neural operator, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.