Shamsulhaq Basir

Shamsulhaq Basir, Inanc Senocak

Several recent works in scientific machine learning have revived interest in the application of neural networks to partial differential equations (PDEs). A popular approach is to aggregate the residual form of the governing PDE and its boundary conditions as soft penalties into a composite objective/loss function for training neural networks, which is commonly referred to as physics-informed neural networks (PINNs). In the present study, we visualize the loss landscapes and distributions of learned parameters and explain the ways this particular formulation of the objective function may hinder or even prevent convergence when dealing with challenging target solutions. We construct a purely data-driven loss function composed of both the boundary loss and the domain loss. Using this data-driven loss function and, separately, a physics-informed loss function, we then train two neural network models with the same architecture. We show that incomparable scales between boundary and domain loss terms are the culprit behind the poor performance. Additionally, we assess the performance of both approaches on two elliptic problems with increasingly complex target solutions. Based on our analysis of their loss landscapes and learned parameter distributions, we observe that a physics-informed neural network with a composite objective function formulation produces highly non-convex loss surfaces that are difficult to optimize and are more prone to the problem of vanishing gradients.

Shamsulhaq Basir, Inanc Senocak

Physics and equality constrained artificial neural networks (PECANN) are grounded in methods of constrained optimization to properly constrain the solution of partial differential equations (PDEs) with their boundary and initial conditions and any high-fidelity data that may be available. To this end, adoption of the augmented Lagrangian method within the PECANN framework is paramount for learning the solution of PDEs without manually balancing the individual loss terms in the objective function used for determining the parameters of the neural network. Generally speaking, ALM combines the merits of the penalty and Lagrange multiplier methods while avoiding the ill conditioning and convergence issues associated singly with these methods . In the present work, we apply our PECANN framework to solve forward and inverse problems that have an expanded and diverse set of constraints. We show that ALM with its conventional formulation to update its penalty parameter and Lagrange multipliers stalls for such challenging problems. To address this issue, we propose an adaptive ALM in which each constraint is assigned a unique penalty parameter that evolve adaptively according to a rule inspired by the adaptive subgradient method. Additionally, we revise our PECANN formulation for improved computational efficiency and savings which allows for mini-batch training. We demonstrate the efficacy of our proposed approach by solving several forward and PDE-constrained inverse problems with noisy data, including simulation of incompressible fluid flows with a primitive-variables formulation of the Navier-Stokes equations up to a Reynolds number of 1000.

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.

Shamsulhaq Basir, Inanc Senocak

We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of solutions across neighboring subdomains, we adopt a generalized Robin-type interface condition, assigning unique Robin parameters to each subdomain. These subdomain-specific Robin parameters are learned to minimize the mismatch on the Robin interface condition, facilitating efficient information exchange during training. Our method is applicable to both the Laplace's and Helmholtz equations. It represents local solutions by an independent neural network model which is trained to minimize the loss on the governing PDE while strictly enforcing boundary and interface conditions through an augmented Lagrangian formalism. A key strength of our method lies in its ability to learn a Robin parameter for each subdomain, thereby enhancing information exchange with its neighboring subdomains. We observe that the learned Robin parameters adapt to the local behavior of the solution, domain partitioning and subdomain location relative to the overall domain. Extensive experiments on forward and inverse problems, including one-way and two-way decompositions with crosspoints, demonstrate the versatility and performance of our proposed approach.

Shamsulhaq Basir, Inanc Senocak

Neural networks can be used to learn the solution of partial differential equations (PDEs) on arbitrary domains without requiring a computational mesh. Common approaches integrate differential operators in training neural networks using a structured loss function. The most common training algorithm for neural networks is backpropagation which relies on the gradient of the loss function with respect to the parameters of the network. In this work, we characterize the difficulty of training neural networks on physics by investigating the impact of differential operators in corrupting the back propagated gradients. Particularly, we show that perturbations present in the output of a neural network model during early stages of training lead to higher levels of noise in a structured loss function that is composed of high-order differential operators. These perturbations consequently corrupt the back-propagated gradients and impede convergence. We mitigate this issue by introducing auxiliary flux parameters to obtain a system of first-order differential equations. We formulate a non-linear unconstrained optimization problem using the augmented Lagrangian method that properly constrains the boundary conditions and adaptively focus on regions of higher gradients that are difficult to learn. We apply our approach to learn the solution of various benchmark PDE problems and demonstrate orders of magnitude improvement over existing approaches.

Shamsulhaq Basir

This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach is the requirement for manual hyperparameter tuning, making it impractical in the absence of validation data or prior knowledge of the solution. Our investigations of the loss landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate. Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence. To address these challenges, we introduce a novel method that bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients. Consequently, we reduce the dimension of the search space and make learning PDEs with non-smooth solutions feasible. Our method also provides a mechanism to focus on complex regions of the domain. Besides, we present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model's prediction, with adaptive and independent learning rates inspired by adaptive subgradient methods. We apply our approach to solve various linear and non-linear PDEs.

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.

Shamsulhaq Basir, Inanc Senocak

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. To address these limitations, we propose a versatile framework based on a constrained optimization problem formulation, where we use the augmented Lagrangian method (ALM) to constrain the solution of a PDE with its boundary conditions and any high-fidelity data that may be available. Our approach is adept at forward and inverse problems with multi-fidelity data fusion. We demonstrate the efficacy and versatility of our physics- and equality-constrained deep-learning framework by applying it to several forward and inverse problems involving multi-dimensional PDEs. Our framework achieves orders of magnitude improvements in accuracy levels in comparison with state-of-the-art physics-informed neural networks.