Physics-Informed Neural Networks for Solving Derivative-Constrained PDEs

• URL: http://arxiv.org/abs/2604.13723v1
• Date: Wed, 15 Apr 2026 10:57:22 GMT
• Title: Physics-Informed Neural Networks for Solving Derivative-Constrained PDEs
• Abstract summary: Physics-Informed Neural Networks (PINNs) recast PDE solving as an optimisation problem in function space by minimising a residual-based objective.<n>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.
• Score: 1.0323063834827415
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Physics-Informed Foresight Pruning for Sparse PINN Solvers of Nonlinear PDEs [8.49003726182843]
  We introduce physics-informed spectrum-aware pruning (PI-SAP) for sparse PirateNet PDE solvers.<n>Experiments show PI-SAP more consistently preserves Gray-Scott residual fidelity and is competitive under aggressive sparsity.<n>Small-batch PINN-NTK diagnostics further show that residual fidelity, solution accuracy, and kernel conditioning are distinct objectives.
  arXiv  Detail & Related papers  (2026-08-26T09:16:24Z)
• Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving [0.29360071145551064]
  Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by embedding physical principles into the learning process.<n>This study proposes a unified hard-soft physics--informed neural network (HSPINN) with adaptive loss weighting.<n>HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs.
  arXiv  Detail & Related papers  (2026-06-22T13:58:56Z)
• PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations [4.7903561901859355]
  We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step.<n>Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase.<n>Across three representative PDE systems, PhysicsCorrect reduces prediction errors by up to 100x while adding negligible inference time.
  arXiv  Detail & Related papers  (2025-07-03T01:22:57Z)
• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs)<n>We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations.<n>We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• Learning differentiable solvers for systems with hard constraints [48.54197776363251]
  We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs) We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.
  arXiv  Detail & Related papers  (2022-07-18T15:11:43Z)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)
• Mitigating Learning Complexity in Physics and Equality Constrained Artificial Neural Networks [0.9137554315375919]
  Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE) In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective function as soft penalties. Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach when applied to different kinds of PDEs.
  arXiv  Detail & Related papers  (2022-06-19T04:12:01Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Physics and Equality Constrained Artificial Neural Networks: Application to Partial Differential Equations [1.370633147306388]
  Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE) Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach. We propose a versatile framework that can tackle both inverse and forward problems.
  arXiv  Detail & Related papers  (2021-09-30T05:55:35Z)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)
• Bayesian neural networks for weak solution of PDEs with uncertainty quantification [3.4773470589069473]
  A new physics-constrained neural network (NN) approach is proposed to solve PDEs without labels. We write the loss function of NNs based on the discretized residual of PDEs through an efficient, convolutional operator-based, and vectorized implementation. We demonstrate the capability and performance of the proposed framework by applying it to steady-state diffusion, linear elasticity, and nonlinear elasticity.
  arXiv  Detail & Related papers  (2021-01-13T04:57:51Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.