Physics-Informed Neural Networks (PINNs)

Joint Stochastic Optimal Control and Stopping in Aquaculture: Finite-Difference and PINN-Based Approaches ArXiv ID: 2510.02910 “View on arXiv” Authors: Kevin Kamm Abstract This paper studies a joint stochastic optimal control and stopping (JCtrlOS) problem motivated by aquaculture operations, where the objective is to maximize farm profit through an optimal feeding strategy and harvesting time under stochastic price dynamics. We introduce a simplified aquaculture model capturing essential biological and economic features, distinguishing between biologically optimal and economically optimal feeding strategies. The problem is formulated as a Hamilton-Jacobi-Bellman variational inequality and corresponding free boundary problem. We develop two numerical solution approaches: First, a finite difference scheme that serves as a benchmark, and second, a Physics-Informed Neural Network (PINN)-based method, combined with a deep optimal stopping (DeepOS) algorithm to improve stopping time accuracy. Numerical experiments demonstrate that while finite differences perform well in medium-dimensional settings, the PINN approach achieves comparable accuracy and is more scalable to higher dimensions where grid-based methods become infeasible. The results confirm that jointly optimizing feeding and harvesting decisions outperforms strategies that neglect either control or stopping. ...

Whack-a-mole Online Learning: Physics-Informed Neural Network for Intraday Implied Volatility Surface ArXiv ID: 2411.02375 “View on arXiv” Authors: Unknown Abstract Calibrating the time-dependent Implied Volatility Surface (IVS) using sparse market data is an essential challenge in computational finance, particularly for real-time applications. This task requires not only fitting market data but also satisfying a specified partial differential equation (PDE) and no-arbitrage conditions modelled by differential inequalities. This paper proposes a novel Physics-Informed Neural Networks (PINNs) approach called Whack-a-mole Online Learning (WamOL) to address this multi-objective optimisation problem. WamOL integrates self-adaptive and auto-balancing processes for each loss term, efficiently reweighting objective functions to ensure smooth surface fitting while adhering to PDE and no-arbitrage constraints and updating for intraday predictions. In our experiments, WamOL demonstrates superior performance in calibrating intraday IVS from uneven and sparse market data, effectively capturing the dynamic evolution of option prices and associated risk profiles. This approach offers an efficient solution for intraday IVS calibration, extending PINNs applications and providing a method for real-time financial modelling. ...