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3 papers in physics.comp-ph
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A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning
math.NAcs.LGcs.NA Suchuan Dong, Yuchuan Zhang · Mar 23, 2026

Physics-informed neural networks typically enforce boundary conditions via penalty terms, leading to approximate satisfaction and training pathologies. This paper proposes a systematic method to enforce Dirichlet, Neumann, and Robin conditions exactly on curved quadrilateral domains using Theory of Functional Connections (TFC) combined with transfinite interpolation. The key innovation is handling compatibility constraints at vertices where mixed boundary conditions meet, particularly when two Neumann/Robin boundaries intersect, by decomposing the problem into a four-step procedure.

We present a systematic method for exactly enforcing Dirichlet, Neumann, and Robin type conditions on general quadrilateral domains with arbitrary curved boundaries. Our method is built upon exact mappings between general quadrilateral domains and the standard domain, and employs a combination of TFC (theory of functional connections) constrained expressions and transfinite interpolations. When Neumann or Robin boundaries are present, especially when two Neumann (or Robin) boundaries meet at a vertex, it is critical to enforce exactly the induced compatibility constraints at the intersection, in order to enforce exactly the imposed conditions on the joining boundaries. We analyze in detail and present constructions for handling the imposed boundary conditions and the induced compatibility constraints for two types of situations: (i) when Neumann (or Robin) boundary only intersects with Dirichlet boundaries, and (ii) when two Neumann (or Robin) boundaries intersect with each other. We describe a four-step procedure to systematically formulate the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on general quadrilateral domains. The method developed herein has been implemented together with the extreme learning machine (ELM) technique we have developed recently for scientific machine learning. Ample numerical experiments are presented with several linear/nonlinear stationary/dynamic problems on a variety of two-dimensional domains with complex boundary geometries. Simulation results demonstrate that the proposed method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries exactly, with the numerical boundary-condition errors at the machine accuracy.
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SPINONet: Scalable Spiking Physics-informed Neural Operator for Computational Mechanics Applications
physics.comp-phcs.LG Shailesh Garg, Luis Mandl, Somdatta Goswami et al. · Mar 23, 2026

Physics-informed neural operators enable rapid surrogate modeling of PDEs but incur substantial energy costs during repeated inference, limiting deployment on edge devices. This paper proposes SPINONet, which embeds Variable Spiking Neurons (VSNs) into the branch network of a separable DeepONet architecture to enable sparse, event-driven computation while preserving continuous coordinate pathways for derivative calculation. The core insight is that structural decoupling—spiking for input encoding and dense differentiability for coordinate encoding—allows physics-informed training without redundant multiply-accumulate operations.

Energy efficiency remains a critical challenge in deploying physics-informed operator learning models for computational mechanics and scientific computing, particularly in power-constrained settings such as edge and embedded devices, where repeated operator evaluations in dense networks incur substantial computational and energy costs. To address this challenge, we introduce the Separable Physics-informed Neuroscience-inspired Operator Network (SPINONet), a neuroscience-inspired framework that reduces redundant computation across repeated evaluations while remaining compatible with physics-informed training. SPINONet incorporates regression-friendly neuroscience-inspired spiking neurons through an architecture-aware design that enables sparse, event-driven computation, improving energy efficiency while preserving the continuous, coordinate-differentiable pathways required for computing spatio-temporal derivatives. We evaluate SPINONet on a range of partial differential equations representative of computational mechanics problems, with spatial, temporal, and parametric dependencies in both time-dependent and steady-state settings, and demonstrate predictive performance comparable to conventional physics-informed operator learning approaches despite the induced sparse communication. In addition, limited data supervision in a hybrid setup is shown to improve performance in challenging regimes where purely physics-informed training may converge to spurious solutions. Finally, we provide an analytical discussion linking architectural components and design choices of SPINONet to reductions in computational load and energy consumption.
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arXivPDF
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Closed-form conditional diffusion models for data assimilation
stat.MLcs.LGphysics.comp-ph Brianna Binder, Assad Oberai · Mar 22, 2026

This paper proposes a training-free conditional diffusion model for Bayesian filtering in data assimilation. Instead of learning the score function via neural networks, the authors leverage kernel density estimation (KDE) to represent the joint distribution of states and measurements, yielding a closed-form expression for the score that enables analytical sampling from the posterior. The method targets nonlinear, non-Gaussian filtering problems where traditional ensemble Kalman filters (EnKF) make restrictive Gaussian approximations and particle filters suffer from weight degeneracy in small-ensemble regimes.

We propose closed-form conditional diffusion models for data assimilation. Diffusion models use data to learn the score function (defined as the gradient of the log-probability density of a data distribution), allowing them to generate new samples from the data distribution by reversing a noise injection process. While it is common to train neural networks to approximate the score function, we leverage the analytical tractability of the score function to assimilate the states of a system with measurements. To enable the efficient evaluation of the score function, we use kernel density estimation to model the joint distribution of the states and their corresponding measurements. The proposed approach also inherits the capability of conditional diffusion models of operating in black-box settings, i.e., the proposed data assimilation approach can accommodate systems and measurement processes without their explicit knowledge. The ability to accommodate black-box systems combined with the superior capabilities of diffusion models in approximating complex, non-Gaussian probability distributions means that the proposed approach offers advantages over many widely used filtering methods. We evaluate the proposed method on nonlinear data assimilation problems based on the Lorenz-63 and Lorenz-96 systems of moderate dimensionality and nonlinear measurement models. Results show the proposed approach outperforms the widely used ensemble Kalman and particle filters when small to moderate ensemble sizes are used.
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arXivPDF