Partial Differential Equation
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Related Articles from SNS
Non-periodic Fourier propagation algorithms for partial differential equations
arXiv:2507.21757v2 Announce Type: replace Abstract: Spectral methods for partial differential equations (PDEs) with non-periodic boundary conditions arising in computational physics often use polynomial expansions on non-uniform grids. Here, we implement a Fourier method that employs fast trigonometric expansions on a uniform grid with non-periodic boundaries using fast discrete sine transforms (DST) or/and discrete cosine transforms (DCT) to solve parabolic PDEs. We implement this method in...
On the training of physics-informed neural operators for solving parametric partial differential equations
Announce Type: new Abstract: Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning. Despite this promise, how to train PINOs efficiently...
DAS-PINNs for high-dimensional partial differential equations: extending deep adaptive sampling to spacetime domains
Announce Type: new Abstract: Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain...
Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations
arXiv:2512.14980v4 Announce Type: replace Abstract: Diffusion models have become a powerful generative prior for solutions of partial differential equations (PDEs). Existing approaches enforce physical constraints either by adding the PDE residuals as loss regularizers or through inference-time adjustments. These methods bias the model away from the true data distribution, which is especially problematic when the governing PDE is misspecified.
On the training of physics-informed neural operators for solving parametric partial differential equations
Announce Type: cross Abstract: Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning. Despite this promise, how to train PINOs efficiently...
Young Measure Based Quantum Linear Programming Algorithms for Nonlinear/Stochastic Multiscale Partial Differential Equations and Homogenization
arXiv:2606.06165v2 Announce Type: replace Abstract: We study quantum algorithms for nonlinear and stochastic homogenization via a Young-measure based linear programming (LP) formulation, which lifts the nonlinear problem to a linear one in higher dimensions by treating the microscale, the gradient, and possible random variables as independent variables, thereby capturing effective macroscopic quantities without directly resolving fine-scale oscillations. The resulting LP is large but...
Practical Aspects on Solving Differential Equations Using Deep Learning: A Primer
arXiv:2408.11266v5 Announce Type: replace Abstract: Deep learning is now common across many scientific fields, including the study of partial differential equations. This article provides a brief, accessible introduction to core deep learning concepts, including neural networks, backpropagation, and the universal approximation theorem. It mainly covers how to use deep learning in solving differential equations.
Numerical Analysis on Backward Stochastic Differential Equations by Finite Transposition Method
arXiv:2606.08731v1 Announce Type: cross Abstract: In this paper, we propose a finite transposition method to solve backward stochastic differential equations (BSDEs, for short). Based on the transposition solution theory for BSDEs, our method offers a promising way of efficiently computing solutions, which can be regarded as an analogous method for BSDEs as the classical finite element method for partial differential equations. Our method has the advantage of easily computable conditional...
Stochastic Differential Equations (SDEs) in NONMEM for Probing Population Pharmacokinetic Model Misspecification: Diagnostic Utility, Practical Considerations, and Future Directions
Population pharmacokinetic (popPK) models are commonly developed using ordinary differential equations (ODEs) to describe deterministic concentration-time profiles, with unexplained variability typically attributed to interindividual variability or residual error. When model misspecification is present, system-level deviations may be absorbed into these conventional variability terms, making the source and magnitude of model inadequacy difficult to assess quantitatively. Stochastic...
Hybrid Neural Ordinary Differential Equations for Data-Efficient Polymerization Modeling with Incomplete Kinetics
Announce Type: new Abstract: Accurate prediction of polymerization dynamics is essential for process design, control, and optimization. Yet, purely mechanistic models require labor-intensive parameterization of partially characterized kinetics, while purely data-driven models demand large, diverse datasets that are costly to obtain, particularly in early-design stages. We propose a hybrid Neural Ordinary Differential Equation (NODE) framework for data-efficient modeling of free-radical...