Physics-Informed Neural Networks for Solving Derivative-Constrained PDEs

TL;DR

This paper introduces Derivative-Constrained PINNs (DC-PINNs), an extension of Physics-Informed Neural Networks that efficiently incorporate derivative-based constraints, improving solution accuracy and physical fidelity for constrained PDE problems.

Contribution

DC-PINNs embed nonlinear derivative constraints into PINNs, enabling more accurate and physically consistent solutions for constrained PDEs with automatic differentiation and adaptive loss balancing.

Findings

DC-PINNs reduce constraint violations compared to baseline PINNs.

They improve physical fidelity in benchmark constrained PDE problems.

Explicit derivative constraints stabilize training and guide solutions.

Abstract

Physics-Informed Neural Networks (PINNs) recast PDE solving as an optimisation problem in function space by minimising a residual-based objective, yet many applications require additional derivative-based relations that are just as fundamental as the governing equations. In this paper, we present Derivative-Constrained PINNs (DC-PINNs), a general framework that treats constrained PDE solving as an optimisation guided by a minimum objective function criterion where the physics resides in the minimum principle. DC-PINNs embed general nonlinear constraints on states and derivatives, e.g., bounds, monotonicity, convexity, incompressibility, computed efficiently via automatic differentiation, and they employ self-adaptive loss balancing to tune the influence of each objective, reducing reliance on manual hyperparameters and problem-specific architectures. DC-PINNs consistently reduce constraint violations and improve physical fidelity versus baseline PINN variants, representative hard-constraint formulations on benchmarks, including heat diffusion with bounds, financial volatilities with arbitrage-free, and fluid flow with vortices shed. Explicitly encoding derivative constraints stabilises training and steers optimisation toward physically admissible minima even when the PDE residual alone is small, providing reliable solutions of constrained PDEs grounded in energy minimum principles.