Qifeng Hu

Qifeng Hu, Shamsulhaq Basir, Inanc Senocak

We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition, designed for physics-informed machine learning of partial differential equations (PDEs) in both forward and inverse contexts. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. Unlike the original PECANN method, which relies solely on initial and boundary conditions to constrain PDEs, our method uses both boundary conditions and the governing PDE to constrain a unique interface loss function for each subdomain. This modification improves the learning of subdomain-specific interface parameters while reducing communication overhead by delaying information exchange between neighboring subdomains. To address the constrained optimization in each subdomain, we apply an augmented Lagrangian method with a conditionally adaptive update strategy, transforming the problem into an unconstrained dual optimization. A distinct advantage of our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations, even in cases with high-wavenumber and complex-valued solutions. Through numerical experiments with up to 64 subdomains, we demonstrate that our method consistently generalizes well as the number of subdomains increases.

Qifeng Hu, Shamsulhaq Basir, Inanc Senocak

We present several advances to the physics and equality constrained artificial neural networks (PECANN) framework that substantially improve its capability to learn solutions of canonical partial differential equations (PDEs). First, we generalize the augmented Lagrangian method (ALM) to support multiple independent penalty parameters, enabling simultaneous enforcement of heterogeneous constraints. Second, we reformulate pointwise constraint enforcement and Lagrange multipliers as expectations over constraint terms, reducing memory overhead and permitting efficient mini-batch training. Third, to address PDEs with oscillatory, multi-scale features, we incorporate Fourier feature mappings and show that a single mapping suffices where multiple mappings or more costly architectures were required in related methods. Fourth, we introduce a time-windowing strategy for long-time evolution in which the terminal state of each window is enforced as an initial-condition constraint for the next, ensuring continuity without discrete time models. Crucially, we propose a conditionally adaptive penalty update (CAPU) strategy for ALM, which preserves the principle that larger constraint violations incur stronger penalties. CAPU accelerates the growth of Lagrange multipliers for selectively challenging constraints, enhancing constraint enforcement during training. We demonstrate the effectiveness of PECANN-CAPU on problems including the transonic rarefaction problem, reversible advection of a passive by a vortex, high-wavenumber Helmholtz and Poisson equations, and inverse identification of spatially varying heat sources. Comparisons with established methods and recent Kolmogorov-Arnold network approaches show that PECANN-CAPU achieves competitive accuracy across all cases. Collectively, these advances improve PECANN's robustness, efficiency, and applicability to demanding problems in scientific computing.

Qifeng Hu, Inanc Senocak

We present a meshless method for the solution of incompressible Navier-Stokes equations in advection-dominated regimes using physics- and equality-constrained artificial neural networks combined with a conditionally adaptive augmented Lagrangian formulation. A single neural network parameterizes both the velocity and pressure fields, and is trained by minimizing the residual of a Poisson's equation for pressure, constrained by the momentum and continuity equations, together with boundary conditions on the velocity field. No boundary conditions are imposed on the pressure field aside from anchoring the pressure at a point to prevent its unbounded development. The training is performed from scratch without labeled data, relying solely on the governing equations and constraints. To enhance accuracy in advection-dominated flows, we employ a single Fourier feature mapping of the input coordinates. The proposed method is demonstrated for the canonical lid-driven cavity flow up to a Reynolds number of 7,500 and for laminar flow over a circular cylinder with inflow-outflow boundary conditions, achieving excellent agreement with benchmark solutions. We further compare the present formulation against alternative objective-function constructions based on different arrangements of the flow equations, thereby highlighting the algorithmic advantages of the proposed formulation centered around the Poisson's equation for pressure.