arXiv subjects

Pratanu Roy

Publications and source records attributed to Pratanu Roy.

9 recordsLinked to original sources

arXiv2026

$\phi-$DeepONet: A Discontinuity Capturing Neural Operator

We present $\phi-$DeepONet, a physics-informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non-smooth behavior. Classical neural operators are based on the universal approximation theorem which assumes that both the operator and the functions it acts on are continuous. However, many scientific and engineering problems involve naturally discontinuous input fields as well as strong and weak discontinuities in the output fields caused by material interfaces. In $\phi$-DeepONet, discontinuities in the input are handled using multiple branch networks, while discontinuities in the output are learned through a nonlinear latent embedding of the interface. This embedding is constructed from a {\it one-hot} representation of the domain decomposition that is combined with the spatial coordinates in a modified trunk network. The outputs of the branch and trunk networks are then combined through a dot product to produce the final solution, which is trained using a physics- and interface-informed loss function. We evaluate $\phi$-DeepONet on several one- and two-dimensional benchmark problems and demonstrate that it delivers accurate and stable predictions even in the presence of strong interface-driven discontinuities.

cs.CE↗

arXiv2026

Hard-constrained Physics-informed Neural Networks for Interface Problems

Physics-informed neural networks (PINNs) have emerged as a flexible framework for solving partial differential equations, but their performance on interface problems remains challenging because continuity and flux conditions are typically imposed through soft penalty terms. The standard soft-constraint formulation leads to imperfect interface enforcement and degraded accuracy near interfaces. We introduce two ansatz-based hard-constrained PINN formulations for interface problems that embed the interface physics into the solution representation and thereby decouple interface enforcement from PDE residual minimization. The first, termed the windowing approach, constructs the trial space from compactly supported windowed subnetworks so that interface continuity and flux balance are satisfied by design. The second, called the buffer approach, augments unrestricted subnetworks with auxiliary buffer functions that enforce boundary and interface constraints at discrete points through a lightweight correction. We study these formulations on one- and two-dimensional elliptic interface benchmarks and compare them with soft-constrained baselines. In one-dimensional problems, hard constraints consistently improve interface fidelity and remove the need for loss-weight tuning; the windowing approach attains very high accuracy (as low as $O(10^{-9})$) on simple structured cases, whereas the buffer approach remains accurate ($\sim O(10^{-5})$) across a wider range of source terms and interface configurations. In two dimensions, the buffer formulation is shown to be more robust because it enforces constraints through a discrete buffer correction, as the windowing construction becomes more sensitive to overlap and corner effects and over-constrains the problem. This positions the buffer method as a straightforward and geometrically flexible approach to complex interface problems.

math.NA↗

arXiv2026

ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learning

Learning solution operators on arbitrary geometries remains a central challenge in scientific machine learning, especially for many-query simulation, physics-informed learning, and evolving geometries requiring accurate, geometry-aware predictions at arbitrary spatial locations. Existing operator-learning methods often rely on structured discretizations, explicit geometry parameterizations, or point-cloud formulations that couple geometric representation with solution-query sampling, limiting flexibility on irregular and non-parameterized domains. We propose the Arbitrary Geometry-encoded Transformer (ArGEnT), a geometry-conditioned attention framework that decouples geometry encoding from query-point evaluation. We develop three variants: self-attention, cross-attention, and hybrid-attention. ArGEnT can be used independently or integrated with neural operators to incorporate non-geometric physical inputs. In the cross-attention variant, geometry is represented by an independently sampled point cloud used to construct keys and values, while arbitrary solution-query points construct queries. This design enables mesh-independent field prediction, reduces sensitivity to query-point distribution, and allows compact geometric representations to condition large-scale solution evaluations. Across benchmarks in fluid dynamics, solid mechanics, and electrochemical systems, ArGEnT consistently improves accuracy and generalization over standard DeepONet, point-cloud-based operator learning, and geometry-aware transformer baselines. In several cases, it reduces prediction errors by more than an order of magnitude while requiring substantially lower training cost than transformer-based baselines. These results demonstrate that decoupled geometry-query attention provides an accurate, scalable, and flexible framework for operator learning on arbitrary geometries.

cs.LG↗

arXiv2024

Scaled-up prediction of steady Navier-Stokes equation with component reduced order modeling

Scaling up new scientific technologies from laboratory to industry often involves demonstrating performance on a larger scale. Computer simulations can accelerate design and predictions in the deployment process, though traditional numerical methods are computationally intractable even for intermediate pilot plant scales. Recently, component reduced order modeling method is developed to tackle this challenge by combining projection reduced order modeling and discontinuous Galerkin domain decomposition. However, while many scientific or engineering applications involve nonlinear physics, this method has been only demonstrated for various linear systems. In this work, the component reduced order modeling method is extended to steady Navier-Stokes flow, with application to general nonlinear physics in view. Large-scale, global domain is decomposed into combination of small-scale unit component. Linear subspaces for flow velocity and pressure are identified via proper orthogonal decomposition over sample snapshots collected at small scale unit component. Velocity bases are augmented with pressure supremizer, in order to satisfy inf-sup condition for stable pressure prediction. Two different nonlinear reduced order modeling methods are employed and compared for efficient evaluation of nonlinear advection: 3rd-order tensor projection operator and empirical quadrature procedure. The proposed method is demonstrated on flow over arrays of five different unit objects, achieving $23$ times faster prediction with less than $4\%$ relative error up to $256$ times larger scale domain than unit components. Furthermore, a numerical experiment with pressure supremizer strongly indicates the need of supremizer for stable pressure prediction. A comparison between tensorial approach and empirical quadrature procedure is performed, which suggests a slight advantage for empirical quadrature procedure.

math.NA↗

arXiv2024

Scalable physics-guided data-driven component model reduction for steady Navier-Stokes flow

Computational physics simulation can be a powerful tool to accelerate industry deployment of new scientific technologies. However, it must address the challenge of computationally tractable, moderately accurate prediction at large industry scales, and training a model without data at such large scales. A recently proposed component reduced order modeling (CROM) tackles this challenge by combining reduced order modeling (ROM) with discontinuous Galerkin domain decomposition (DG-DD). While it can build a component ROM at small scales that can be assembled into a large scale system, its application is limited to linear physics equations. In this work, we extend CROM to nonlinear steady Navier-Stokes flow equation. Nonlinear advection term is evaluated via tensorial approach or empirical quadrature procedure. Application to flow past an array of objects at moderate Reynolds number demonstrates $\sim23.7$ times faster solutions with a relative error of $\sim 2.3\%$, even at scales $256$ times larger than the original problem.

math.NA↗

arXiv2024

Physics-informed Neural Networks for Heterogeneous Poroelastic Media

This study presents a novel physics-informed neural network (PINN) framework for modeling poroelasticity in heterogeneous media with material interfaces. The approach introduces a composite neural network (CoNN) where separate neural networks predict displacement and pressure variables for each material. While sharing identical activation functions, these networks are independently trained for all other parameters. To address challenges posed by heterogeneous material interfaces, the CoNN is integrated with the Interface-PINNs or I-PINNs framework (Sarma et al. 2024, https://dx.doi.org/10.1016/j.cma.2024.117135), allowing different activation functions across material interfaces. This ensures accurate approximation of discontinuous solution fields and gradients. Performance and accuracy of this combined architecture were evaluated against the conventional PINNs approach, a single neural network (SNN) architecture, and the eXtended PINNs (XPINNs) framework through two one-dimensional benchmark examples with discontinuous material properties. The results show that the proposed CoNN with I-PINNs architecture achieves an RMSE that is two orders of magnitude better than the conventional PINNs approach and is at least 40 times faster than the SNN framework. Compared to XPINNs, the proposed method achieves an RMSE at least one order of magnitude better and is 40% faster.

eess.SY↗

arXiv2024

Adaptive Interface-PINNs (AdaI-PINNs): An Efficient Physics-informed Neural Networks Framework for Interface Problems

We present an efficient physics-informed neural networks (PINNs) framework, termed Adaptive Interface-PINNs (AdaI-PINNs), to improve the modeling of interface problems with discontinuous coefficients and/or interfacial jumps. This framework is an enhanced version of its predecessor, Interface PINNs or I-PINNs (Sarma et al.; https://dx.doi.org/10.2139/ssrn.4766623), which involves domain decomposition and assignment of different predefined activation functions to the neural networks in each subdomain across a sharp interface, while keeping all other parameters of the neural networks identical. In AdaI-PINNs, the activation functions vary solely in their slopes, which are trained along with the other parameters of the neural networks. This makes the AdaI-PINNs framework fully automated without requiring preset activation functions. Comparative studies on one-dimensional, two-dimensional, and three-dimensional benchmark elliptic interface problems reveal that AdaI-PINNs outperform I-PINNs, reducing computational costs by 2-6 times while producing similar or better accuracy.

cs.LG↗

arXiv2024

Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networks

The use of deep learning methods in scientific computing represents a potential paradigm shift in engineering problem solving. One of the most prominent developments is Physics-Informed Neural Networks (PINNs), in which neural networks are trained to satisfy partial differential equations (PDEs). While this method shows promise, the standard version has been shown to struggle in accurately predicting the dynamic behavior of time-dependent problems. To address this challenge, methods have been proposed that decompose the time domain into multiple segments, employing a distinct neural network in each segment and directly incorporating continuity between them in the loss function of the minimization problem. In this work we introduce a method to exactly enforce continuity between successive time segments via a solution ansatz. This hard constrained sequential PINN (HCS-PINN) method is simple to implement and eliminates the need for any loss terms associated with temporal continuity. The method is tested for a number of benchmark problems involving both linear and non-linear PDEs. Examples include various first order time dependent problems in which traditional PINNs struggle, namely advection, Allen-Cahn, and Korteweg-de Vries equations. Furthermore, second and third order time-dependent problems are demonstrated via wave and Jerky dynamics examples, respectively. Notably, the Jerky dynamics problem is chaotic, making the problem especially sensitive to temporal accuracy. The numerical experiments conducted with the proposed method demonstrated superior convergence and accuracy over both traditional PINNs and the soft-constrained counterparts.

cs.LG↗

arXiv2023

Train Small, Model Big: Scalable Physics Simulators via Reduced Order Modeling and Domain Decomposition

Numerous cutting-edge scientific technologies originate at the laboratory scale, but transitioning them to practical industry applications is a formidable challenge. Traditional pilot projects at intermediate scales are costly and time-consuming. An alternative, the E-pilot, relies on high-fidelity numerical simulations, but even these simulations can be computationally prohibitive at larger scales. To overcome these limitations, we propose a scalable, physics-constrained reduced order model (ROM) method. ROM identifies critical physics modes from small-scale unit components, projecting governing equations onto these modes to create a reduced model that retains essential physics details. We also employ Discontinuous Galerkin Domain Decomposition (DG-DD) to apply ROM to unit components and interfaces, enabling the construction of large-scale global systems without data at such large scales. This method is demonstrated on the Poisson and Stokes flow equations, showing that it can solve equations about $15 - 40$ times faster with only $\sim$ $1\%$ relative error. Furthermore, ROM takes one order of magnitude less memory than the full order model, enabling larger scale predictions at a given memory limitation.

cs.CE↗