Physics-informed neural networks for solving derivative-constrained partial differential equations

Physics-informed neural networks (PINNs) recast partial differential equations (PDEs) solving as an optimization problem in function space by minimizing 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 optimization 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 stabilizes training and steers optimization toward physically admissible minima even when the PDE residual alone is small, providing reliable solutions of constrained PDEs grounded in energy minimum principles.