Harnessing physics-informed operators for high-dimensional reliability analysis problems

TL;DR

This work investigates the efficacy of the recently developed physics-informed wavelet neural operator in solving reliability analysis problems and investigates the possibility of using physics-informed operator for solving high-dimensional reliability analysis problems, while bypassing the need for any simulation.

Abstract

Reliability analysis is a formidable task, particularly in systems with a large number of stochastic parameters. Conventional methods for quantifying reliability often rely on extensive simulations or experimental data, which can be costly and time-consuming, especially when dealing with systems governed by complex physical laws which necessitates computationally intensive numerical methods such as finite element or finite volume techniques. On the other hand, surrogate-based methods offer an efficient alternative for computing reliability by approximating the underlying model from limited data. Neural operators have recently emerged as effective surrogates for modelling physical systems governed by partial differential equations. These operators can learn solutions to PDEs for varying inputs and parameters. Here, we investigate the efficacy of the recently developed physics-informed wavelet neural operator in solving reliability analysis problems. In particular, we investigate the possibility of using physics-informed operator for solving high-dimensional reliability analysis problems, while bypassing the need for any simulation. Through four numerical examples, we illustrate that physics-informed operator can seamlessly solve high-dimensional reliability analysis problems with reasonable accuracy, while eliminating the need for running expensive simulations.

Figures (17)

• Figure 1: The Proposed Physics-informed Operator (PIO) for reliability analysis. Here, we propose physics-informed WNO (PIWNO) as the PIO, where inputs are initially lifted to a high-dimensional latent space, where they undergo iterative processes. These iterations are represented using wavelet kernel integration blocks, which consist of a kernel integration network that learns the integration kernel and a linear transformation network. The latent inputs are transformed into the space-frequency localised domain using wavelets. The outputs of the integration and the integral constants are then combined, and a nonlinear activation is applied. Output from the ultimate integral layer is down-lifted to obtain the final output, which is obtained as the solution for the underlying PDE. The solutions are constrained to satisfy the given PDEs, boundary conditions (BC), and initial conditions (IC). To enforce the PDE constraints, spatial derivatives are computed using a stochastic projection-based gradient estimation scheme.
• Figure 2: A diagrammatic representation grid point (red dot) and neighbourhood region utilized compute stochastic projection based gradient, where the black dots denote the neighbourhood collocation points
• Figure 3: The results for the Diffusion-reaction system comprised of source functions, ground truth solutions, predictions, and error plots demonstrated for using 2 different unseen sample instances. The PIO effectively maps the initial condition to the corresponding solution $u(x,t)$ over the domain, with a spatiotemporal resolution of $81 \times 81$.
• Figure 4: PDF plots of failure time obtained by MCS and PIO trained with varying number of source functions for the diffusion-reaction system
• Figure 5: Prediction results of the diffusion-reaction system for the failure probability ($P_f$) and the reliability index ($\beta$) with unceasing limit state threshold, obtained by PIO in comparison with the results of MCS, FORM, SORM and data-driven WNOs (trained with a number of samples, $N_s = 300$ and $N_s = 600$)