Physics-Informed Neural Networks for Portfolio Optimisation and Algorithmic Trading: A Reproducible Pipeline with an Indian-Market Backtest

Abstract

Deep learning has entered algorithmic trading largely as data-driven pattern fitting, which markets punish when regimes change. Physics-informed neural networks offer a different discipline: they embed governing equations directly in the training loss, so the model respects the dynamics even where data are scarce. This paper applies that discipline to portfolio optimisation. The reference problem is Merton’s continuous-time allocation between a risky index and a  risk-free asset, whose optimal policy solves a Hamilton-Jacobi-Bellman equation with, for constant relative risk aversion, a closed-form solution. We train a from-scratch physics-informed network to solve the reduced equation with no market data at all, only the equation, and validate that it recovers both the analytic value function, to a relative error below two times ten to the minus four, and the closed-form Merton allocation exactly. Two guarantees support the method: the closed-form policy itself, and an a-posteriori bound showing that the value-function error is controlled by the physics residual. We then backtest the allocation on a seeded simulator calibrated to plausible Indian-market parameters, net of Indian transaction costs, against buy and-hold, a fixed mix, an unrealisable oracle, and a tradable estimation rule. The lesson is deliberately sober: the physics supplies the correct allocation rule, a constant policy on a stable long-run premium is competitive with a sensible fixed mix and far gentler on drawdown than buy-and-hold, and the value that remains is almost entirely in estimating the inputs, where an oracle would reach a Sharpe ratio of 1.10 but a naive adaptive estimator falls to 0.34 after estimation error and turnover costs. Embedding the governing equations makes the model data-efficient and explainable; it does not repeal the estimation problem, and honest system design should place its effort there. The backtest uses a market simulation, not real exchange
data.