# Beyond Time Induction: Structured Conditional Neural Solvers for Parametric Stochastic-Volatility Pricing

arXiv: https://arxiv.org/abs/2610.19254 · HTML: https://mlsanity.com/paper/2610.19254/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 23 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 16 of 16)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [Conditionally Parameterized, Discretization-Aware Neural Networks for Mesh-Based Modeling of Physical Systems](https://mlsanity.com/paper/2109.09510/) (2021)
- [Differential Machine Learning](https://mlsanity.com/paper/2005.02347/) (2020)
- [TINNs: Time-Induced Neural Networks for Solving Time-Dependent PDEs](https://mlsanity.com/paper/2601.20361/) (2026)
- [DeepSVM: Learning Stochastic Volatility Models with Physics-Informed Deep Operator Networks](https://mlsanity.com/paper/2512.07162/) (2025)
- [Neural Operator Processes for Probabilistic Operator Learning under Partial Observations](https://mlsanity.com/paper/2606.22946/) (2026)

## Abstract (arXiv)

Physics-informed neural networks provide a flexible framework for learning parametric PDE solution families without labeled data, yet preserving accurate derivatives across heterogeneous regimes remains challenging. Structured neural solvers offer compact representations, but restrictive conditional parameterizations can limit adaptation across the solution family. We introduce BeTINN, a structured conditional architecture extending time-induced modulation through multi-directional time-parameter conditioning and a joint residual pathway. We evaluate BeTINN on the Heston stochastic-volatility PDE, a demanding parametric problem with mixed derivatives and economically relevant first- and second-order sensitivities. BeTINN improves joint accuracy across normalized option value and key sensitivity metrics relative to our baselines. Controlled ablations characterize the roles of its architectural components. We further find that validation physics loss is informative but does not fully determine differential fidelity, and that accurate solution values can coexist with localized derivative errors. These results support structured conditional adaptation and motivate derivative-aware evaluation when sensitivities are downstream quantities of interest.