Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints

• URL: http://arxiv.org/abs/2507.08124v1
• Date: Thu, 10 Jul 2025 19:24:53 GMT
• Title: Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints
• Abstract summary: KKT-Hardnet is a PINN architecture that enforces both linear and nonlinear equality and inequality constraints up to machine precision.<n>We apply KKT-Hardnet on both test problems and a real-world chemical process simulation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Traditional physics-informed neural networks (PINNs) do not guarantee strict constraint satisfaction. This is problematic in engineering systems where minor violations of governing laws can significantly degrade the reliability and consistency of model predictions. In this work, we develop KKT-Hardnet, a PINN architecture that enforces both linear and nonlinear equality and inequality constraints up to machine precision. It leverages a projection onto the feasible region through solving Karush-Kuhn-Tucker (KKT) conditions of a distance minimization problem. Furthermore, we reformulate the nonlinear KKT conditions using log-exponential transformation to construct a general sparse system with only linear and exponential terms, thereby making the projection differentiable. We apply KKT-Hardnet on both test problems and a real-world chemical process simulation. Compared to multilayer perceptrons and PINNs, KKT-Hardnet achieves higher accuracy and strict constraint satisfaction. This approach allows the integration of domain knowledge into machine learning towards reliable hybrid modeling of complex systems.

Related papers

• Linearized PINN with pretrained nonlinear layers [48.01541150431857]
  lPINN is a reduced-order neural basis method for forward and inverse differential equations.<n>We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation.
  arXiv  Detail & Related papers  (2026-09-14T02:18:00Z)
• INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities [41.99844472131922]
  We propose a new weak-form Physics-Informed Neural Network approach (named INI-VPINN)<n>It naturally incorporates Neumann boundary and interface conditions into the variational formulation.<n>The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann-Dirichlet boundary conditions using neural networks.
  arXiv  Detail & Related papers  (2026-06-16T15:06:15Z)
• PL-KKT-hPINN: Enforcing Nonlinear Equality Constraints on Neural Networks via Piecewise-Linear Projection [0.0]
  We propose a framework, called piecewise-linear Karush--Kuhn--Tucker hard-constrained PINNs.<n>PL-KKT-hPINN strictly enforces nonlinear equality constraints through piecewise-linear projection.<n>Results show that PL-KKT-hPINN preserves predictive accuracy comparable to that of a standard neural network.
  arXiv  Detail & Related papers  (2026-06-09T10:38:06Z)
• Learning partially observed systems with neural Hamiltonian ordinary differential equations [41.99844472131922]
  We present neural Hamiltonian ordinary differential equations (NHODE) to learn partially observed dynamical systems from data.<n>The framework is evaluated on systems of increasing complexity, from linear and nonlinear mass-spring systems to the chaotic three-body problem.
  arXiv  Detail & Related papers  (2026-05-22T11:18:39Z)
• Comparing Physics-Informed and Neural ODE Approaches for Modeling Nonlinear Biological Systems: A Case Study Based on the Morris-Lecar Model [0.0]
  This study systematically evaluates the performance of PINNs and NODEs across three canonical bifurcation regimes.<n>Results indicate that PINNs tend to achieve higher accuracy and robustness in scenarios involving stiffness or sensitive bifurcations.
  arXiv  Detail & Related papers  (2026-03-27T18:53:53Z)
• Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations [14.314318817152165]
  We develop a novel framework to solve the incompressible magnetohydrodynamic (MHD) equations.<n>It preserves the strong nonlinearity and dual divergence-free constraints.<n>It achieves higher accuracy, faster convergence, and exact enforcement of divergence-free constraints.
  arXiv  Detail & Related papers  (2026-03-01T13:42:28Z)
• Solving nonlinear subsonic compressible flow in infinite domain via multi-stage neural networks [1.1317872970491056]
  In aerodynamics, accurately modeling subsonic compressible flow over airfoils is critical for aircraft design.<n>Traditional approaches often rely on linearized equations or finite, truncated domains, which introduce non-negligible errors and limit applicability in real-world scenarios.<n>We propose a novel framework utilizing Physics-Informed Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded domain.
  arXiv  Detail & Related papers  (2026-01-01T13:46:01Z)
• Solved in Unit Domain: JacobiNet for Differentiable Coordinate Transformations [8.832884051186426]
  JacobiNet is a neural network-based computation method that learns continuous, differentiable mappings from supervised point pairs.<n>It effectively addresses normalization challenges, facilitates hard constraints of boundary conditions, and mitigates the long-standing imbalance among loss terms.<n>It demonstrates significant improvements, reducing the relative L2 error from 0.287-0.637 to 0.013-0.039, achieving an average accuracy improvement of 18.3*.
  arXiv  Detail & Related papers  (2025-08-04T15:45:03Z)
• LaPON: A Lagrange's-mean-value-theorem-inspired operator network for solving PDEs and its application on NSE [8.014720523981385]
  We propose LaPON, an operator network inspired by the Lagrange's mean value theorem.<n>It embeds prior knowledge directly into the neural network architecture instead of the loss function.<n>LaPON provides a scalable and reliable solution for high-fidelity fluid dynamics simulation.
  arXiv  Detail & Related papers  (2025-05-18T10:45:17Z)
• Hamiltonian Neural Networks approach to fuzzball geodesics [39.58317527488534]
  Hamiltonian Neural Networks (HNNs) are tools that minimize a loss function to solve Hamilton equations of motion.<n>In this work, we implement several HNNs trained to solve, with high accuracy, the Hamilton equations for a massless probe moving inside a smooth and horizonless geometry known as D1-D5 circular fuzzball.
  arXiv  Detail & Related papers  (2025-02-28T09:25:49Z)
• ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection [0.0]
  We introduce ENFORCE, a neural network architecture that uses an adaptive projection module (AdaNP) to enforce nonlinear equality constraints in the predictions.<n>We prove that our projection mapping is 1-Lipschitz, making it well-suited for stable training.<n>The predictions of our new architecture satisfy $N_C$ equality constraints that are nonlinear in both the inputs and outputs of the neural network.
  arXiv  Detail & Related papers  (2025-02-10T18:52:22Z)
• Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems [41.94295877935867]
  numerical simulation and optimization of technical systems described by partial differential equations is expensive.<n>A comparatively new approach in this context is to combine the good approximation properties of neural networks with the classical finite element method.<n>In this paper, we extend this approach to saddle-point and non-linear fluid dynamics problems, respectively.
  arXiv  Detail & Related papers  (2024-09-06T07:17:01Z)
• Scaling physics-informed hard constraints with mixture-of-experts [0.0]
  We develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE) MoE imposes the constraint over smaller domains, each of which is solved by an "expert" through differentiable optimization. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting.
  arXiv  Detail & Related papers  (2024-02-20T22:45:00Z)
• Physics-Informed Neural Networks with Hard Linear Equality Constraints [9.101849365688905]
  This work proposes a novel physics-informed neural network, KKT-hPINN, which rigorously guarantees hard linear equality constraints. Experiments on Aspen models of a stirred-tank reactor unit, an extractive distillation subsystem, and a chemical plant demonstrate that this model can further enhance the prediction accuracy.
  arXiv  Detail & Related papers  (2024-02-11T17:40:26Z)
• Neural Fields with Hard Constraints of Arbitrary Differential Order [61.49418682745144]
  We develop a series of approaches for enforcing hard constraints on neural fields. The constraints can be specified as a linear operator applied to the neural field and its derivatives. Our approaches are demonstrated in a wide range of real-world applications.
  arXiv  Detail & Related papers  (2023-06-15T08:33:52Z)
• Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching [55.28394191394675]
  We develop an adaptive inexact Newton method for equality-constrained nonlinear, nonIBS optimization problems. We demonstrate the superior performance of our method on benchmark nonlinear problems, constrained logistic regression with data from LVM, and a PDE-constrained problem.
  arXiv  Detail & Related papers  (2023-05-28T06:33:37Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• 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)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)
• Robust Implicit Networks via Non-Euclidean Contractions [63.91638306025768]
  Implicit neural networks show improved accuracy and significant reduction in memory consumption. They can suffer from ill-posedness and convergence instability. This paper provides a new framework to design well-posed and robust implicit neural networks.
  arXiv  Detail & Related papers  (2021-06-06T18:05:02Z)
• Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints [49.66841118264278]
  We introduce a series of challenging chaotic and extended-body systems to push the limits of current approaches. Our experiments show that Cartesian coordinates with explicit constraints lead to a 100x improvement in accuracy and data efficiency.
  arXiv  Detail & Related papers  (2020-10-26T13:35:16Z)

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.