arXiv subjects

Xuhui Meng

Publications and source records attributed to Xuhui Meng.

At least 19 recordsLinked to original sources

arXiv2026

Monte Carlo Physics-informed Neural Networks for Inverse Multiscale Heat Conduction Problems via the Phonon Boltzmann Transport Equation

Inferring thermal fields and thermophysical properties from limited measurements is a fundamental challenge in micro- and nanoscale heat conduction, where the classical Fourier law breaks down and the phonon Boltzmann transport equation (BTE) is needed to capture non-diffusive transport effects. In this work, we extend Monte Carlo physics-informed neural networks (MC-PINNs), originally developed for forward phonon BTE problems [J. Comput. Phys. 542, 114364, 2025], to inverse multiscale heat conduction problems. Two representative classes of inverse problems are considered: (i) reconstructing the full thermal field from sparse interior temperature measurements when boundary conditions are unknown, and (ii) simultaneously inferring the unknown relaxation time together with the thermal field. Problem-specific MC-PINN architectures and training strategies are designed for each class. The mesh-free Monte Carlo sampling strategy enables a unified treatment across diffusive, transitional, and ballistic transport regimes without requiring a priori knowledge of the relaxation time. The proposed method is evaluated on quasi-one-dimensional, quasi-two-dimensional, and three-dimensional benchmark problems covering a wide range of Knudsen numbers, as well as on a realistic 3D fin field-effect transistor (FinFET) structure. Results demonstrate that MC-PINNs consistently outperform purely data-driven deep neural networks, particularly in the sparse-data regime, and can accurately infer spatially uniform relaxation times. For spatially varying relaxation times, the inferred distributions capture the dominant thermal response, and numerical simulations using the recovered parameters reproduce the macroscopic fields with good accuracy. These findings establish MC-PINNs as an effective and physically consistent framework for inverse thermal analysis at micro- and nanoscales.

physics.comp-ph↗

arXiv2026

RepNN: Tackling spectral bias in deep neural networks via parameter reparameterization

Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with activation ReLU or tanh designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one- and four-dimensional function approximations, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features with slightly additional computational cost. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.

cs.LG↗

arXiv2026

Flow-based generative models for amortized Bayesian inference in regression and inverse PDE problems

Bayesian inference provides a principled framework for uncertainty quantification in scientific machine learning. However, conventional Bayesian approaches usually require solving a new inference problem for each observation set, causing substantial computational costs that hinder real-time applications like online monitoring and digital twins. Furthermore, inferring over infinite-dimensional function spaces with varying observation sets poses major challenges for existing amortized inference methods. In this work, we propose Flow-ABI, a flow-based generative framework for amortized Bayesian inference in regression and inverse partial differential equation (PDE) problems. It consists of two components: (i) a functional prior model that learns expressive priors from historical data and physical knowledge through flow matching, and (ii) a set-conditioned functional posterior sampler mapping observation sets to functional posterior distributions. The learned posterior model naturally accommodates varying, permutation-invariant observation sets, and generalizes across different observation discretizations. Once trained, Flow-ABI enables near-real-time posterior sampling for previously unseen observations without retraining or iterative optimization. The proposed methodology can be seamlessly integrated with a wide class of scientific machine learning frameworks, including physics-informed neural networks and neural operators, for uncertainty-aware inverse PDE modeling. Experiments demonstrate that Flow-ABI accurately captures both Gaussian and non-Gaussian posterior distributions while achieving over two-order-of-magnitude speedups relative to the gold-standard Bayesian inference method, Hamiltonian Monte Carlo. These results show Flow-ABI is an effective, scalable, and computationally efficient framework for uncertainty quantification in scientific machine learning.

physics.comp-ph↗

arXiv2025

Self-supervised neural operator for solving partial differential equations

Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high-fidelity data from numerical solvers, limiting applications in complex systems. We propose a self-supervised neural operator (SNO) that generates accurate and diverse training data on the fly without numerical solvers. SNO consists of three parts: a physics-informed sampler (PI-sampler) based on Bayesian PINNs for efficient data generation, a function encoder (FE) for compact input-output representations, and an encoder-only Transformer for operator learning, mapping boundary/initial conditions, source terms, and geometries to PDE solutions. We validate SNO on 1D steady/unsteady nonlinear reaction-diffusion equations, a 2D nonlinear PDE with varying geometries, and vortex-induced vibration of a flexible cylinder in fluid dynamics. SNO achieves high accuracy in all cases, and lightweight finetuning (O(100) trainable variables) further improves predictions with only a few hundred steps. This work provides a new route toward pretrained foundation models as efficient PDE surrogates.

physics.comp-ph↗

arXiv2025

A Machine Learning Framework for Climate-Resilient Insurance and Real Estate Decisions

Extreme weather events increasingly threaten the insurance and real estate industries, creating conflicts between profitability and homeowner burdens. To address this, we propose the SSC-Insurance Model, which integrates SMOTE, SVM, and C-D-C algorithms to evaluate weather impacts on policies and investments. Our model achieves 88.3% accuracy in Zhejiang and 79.6% in Ireland, identifying a critical threshold (43% weather increase) for insurance viability. Additionally, we develop the TOA-Preservation Model using TOPSIS-ORM and AHP to prioritize building protection, with cultural value scoring highest (weight: 0.3383). Case studies on Nanxun Ancient Town show a 65.32% insurability probability and a protection score of 0.512. This work provides actionable tools for insurers, developers, and policymakers to manage climate risks sustainably.

cs.CE↗

arXiv2025

Scalable physics-informed deep generative model for solving forward and inverse stochastic differential equations

Physics-informed deep learning approaches have been developed to solve forward and inverse stochastic differential equation (SDE) problems with high-dimensional stochastic space. However, the existing deep learning models have difficulties solving SDEs with high-dimensional spatial space. In the present study, we propose a scalable physics-informed deep generative model (sPI-GeM), which is capable of solving SDE problems with both high-dimensional stochastic and spatial space. The sPI-GeM consists of two deep learning models, i.e., (1) physics-informed basis networks (PI-BasisNet), which are used to learn the basis functions as well as the coefficients given data on a certain stochastic process or random field, and (2) physics-informed deep generative model (PI-GeM), which learns the distribution over the coefficients obtained from the PI-BasisNet. The new samples for the learned stochastic process can then be obtained using the inner product between the output of the generator and the basis functions from the trained PI-BasisNet. The sPI-GeM addresses the scalability in the spatial space in a similar way as in the widely used dimensionality reduction technique, i.e., principal component analysis (PCA). A series of numerical experiments, including approximation of Gaussian and non-Gaussian stochastic processes, forward and inverse SDE problems, are performed to demonstrate the accuracy of the proposed model. Furthermore, we also show the scalability of the sPI-GeM in both the stochastic and spatial space using an example of a forward SDE problem with 38- and 20-dimension stochastic and spatial space, respectively.

physics.comp-ph↗

arXiv2024

NeDF: neural deflection fields for sparse-view tomographic background oriented Schlieren

Three-dimensional (3D) density-varying turbulent flows are widely encountered in high-speed aerodynamics, combustion, and heterogeneous mixing processes. Multi-camera-based tomographic background-oriented Schlieren (TBOS) has emerged as a powerful technique for revealing 3D flow density structures. However, dozens of cameras are typically required to obtain high-quality reconstructed density fields. Limited by the number of available optical windows and confined space in the harsh experimental environments, TBOS with only sparse views and limited viewing angles often becomes the necessary choice practically, rendering the inverse problem for TBOS reconstruction severely ill-posed and resulting in degraded tomography quality. In this study, we propose a novel TBOS reconstruction method, neural deflection field (NeDF), utilizing deep neural networks (DNNs) to represent the density gradient fields without using any pretrained neural network models. Particularly, state-of-the-art positional encoding techniques and hierarchical sampling strategies are incorporated to capture the density structures of high spatial frequencies. Required background images for TBOS reconstructions are synthesized based on a high-fidelity nonlinear ray-tracing method with the ground truth flows from conducting LES simulations on premixed turbulent flames. Owing to these synthesized BOS images, the superiority of the proposed method is quantitatively verified compared to the classical TBOS reconstruction methods, and the specific contributions from the position encoding and the hierarchical sampling strategy are also elucidated.

physics.flu-dyn↗

arXiv2024

Monte Carlo Physics-informed neural networks for multiscale heat conduction via phonon Boltzmann transport equation

The phonon Boltzmann transport equation (BTE) is widely used for describing multiscale heat conduction (from nm to $\mu$m or mm) in solid materials. Developing numerical approaches to solve this equation is challenging since it is a 7-dimensional integral-differential equation. In this work, we propose Monte Carlo physics-informed neural networks (MC-PINNs), which do not suffer from the "curse of dimensionality", to solve the phonon BTE to model the multiscale heat conduction in solid materials. MC-PINNs use a deep neural network to approximate the solution to the BTE, and encode the BTE as well as the corresponding boundary/initial conditions using the automatic differentiation. In addition, we propose a novel two-step sampling approach to address inefficiency and inaccuracy issues in the widely used sampling methods in PINNs. In particular, we first randomly sample a certain number of points in the temporal-spatial space (Step I), and then draw another number of points randomly in the solid angular space (Step II). The training points at each step are constructed based on the data drawn from the above two steps using the tensor product. The two-step sampling strategy enables MC-PINNs (1) to model the heat conduction from ballistic to diffusive regimes, and (2) is more memory-efficient compared to conventional numerical solvers or existing PINNs for BTE. A series of numerical examples including quasi-one-dimensional (quasi-1D) steady/unsteady heat conduction in a film, and the heat conduction in a quasi-two- and three-dimensional square domains, are conducted to justify the effectiveness of the MC-PINNs for heat conduction spanning diffusive and ballistic regimes. Finally, we compare the computational time and memory usage of the MC-PINNs and one of the state-of-the-art numerical methods to demonstrate the potential of the MC-PINNs for large scale problems in real-world applications.

physics.comp-ph↗

arXiv2023

Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators

Uncertainty quantification (UQ) in scientific machine learning (SciML) becomes increasingly critical as neural networks (NNs) are being widely adopted in addressing complex problems across various scientific disciplines. Representative SciML models are physics-informed neural networks (PINNs) and neural operators (NOs). While UQ in SciML has been increasingly investigated in recent years, very few works have focused on addressing the uncertainty caused by the noisy inputs, such as spatial-temporal coordinates in PINNs and input functions in NOs. The presence of noise in the inputs of the models can pose significantly more challenges compared to noise in the outputs of the models, primarily due to the inherent nonlinearity of most SciML algorithms. As a result, UQ for noisy inputs becomes a crucial factor for reliable and trustworthy deployment of these models in applications involving physical knowledge. To this end, we introduce a Bayesian approach to quantify uncertainty arising from noisy inputs-outputs in PINNs and NOs. We show that this approach can be seamlessly integrated into PINNs and NOs, when they are employed to encode the physical information. PINNs incorporate physics by including physics-informed terms via automatic differentiation, either in the loss function or the likelihood, and often take as input the spatial-temporal coordinate. Therefore, the present method equips PINNs with the capability to address problems where the observed coordinate is subject to noise. On the other hand, pretrained NOs are also commonly employed as equation-free surrogates in solving differential equations and Bayesian inverse problems, in which they take functions as inputs. The proposed approach enables them to handle noisy measurements for both input and output functions with UQ.

cs.LG↗

arXiv2023

Correcting model misspecification in physics-informed neural networks (PINNs)

Data-driven discovery of governing equations in computational science has emerged as a new paradigm for obtaining accurate physical models and as a possible alternative to theoretical derivations. The recently developed physics-informed neural networks (PINNs) have also been employed to learn governing equations given data across diverse scientific disciplines. Despite the effectiveness of PINNs for discovering governing equations, the physical models encoded in PINNs may be misspecified in complex systems as some of the physical processes may not be fully understood, leading to the poor accuracy of PINN predictions. In this work, we present a general approach to correct the misspecified physical models in PINNs for discovering governing equations, given some sparse and/or noisy data. Specifically, we first encode the assumed physical models, which may be misspecified, then employ other deep neural networks (DNNs) to model the discrepancy between the imperfect models and the observational data. Due to the expressivity of DNNs, the proposed method is capable of reducing the computational errors caused by the model misspecification and thus enables the applications of PINNs in complex systems where the physical processes are not exactly known. Furthermore, we utilize the Bayesian PINNs (B-PINNs) and/or ensemble PINNs to quantify uncertainties arising from noisy and/or gappy data in the discovered governing equations. A series of numerical examples including non-Newtonian channel and cavity flows demonstrate that the added DNNs are capable of correcting the model misspecification in PINNs and thus reduce the discrepancy between the physical models and the observational data. We envision that the proposed approach will extend the applications of PINNs for discovering governing equations in problems where the physico-chemical or biological processes are not well understood.

cs.LG↗

arXiv2023

Solution multiplicity and effects of data and eddy viscosity on Navier-Stokes solutions inferred by physics-informed neural networks

Physics-informed neural networks (PINNs) have emerged as a new simulation paradigm for fluid flows and are especially effective for inverse and hybrid problems. However, vanilla PINNs often fail in forward problems, especially at high Reynolds (Re) number flows. Herein, we study systematically the classical lid-driven cavity flow at $Re=2,000$, $3,000$ and $5,000$. We observe that vanilla PINNs obtain two classes of solutions, one class that agrees with direct numerical simulations (DNS), and another that is an unstable solution to the Navier-Stokes equations and not physically realizable. We attribute this solution multiplicity to singularities and unbounded vorticity, and we propose regularization methods that restore a unique solution within 1\% difference from the DNS solution. In particular, we introduce a parameterized entropy-viscosity method as artificial eddy viscosity and identify suitable parameters that drive the PINNs solution towards the DNS solution. Furthermore, we solve the inverse problem by subsampling the DNS solution, and identify a new eddy viscosity distribution that leads to velocity and pressure fields almost identical to their DNS counterparts. Surprisingly, a single measurement at a random point suffices to obtain a unique PINNs DNS-like solution even without artificial viscosity, which suggests possible pathways in simulating high Reynolds number turbulent flows using vanilla PINNs.

physics.flu-dyn↗

arXiv2023

Physics-informed neural networks for predicting gas flow dynamics and unknown parameters in diesel engines

This paper presents a physics-informed neural network (PINN) approach for monitoring the health of diesel engines. The aim is to evaluate the engine dynamics, identify unknown parameters in a "mean value" model, and anticipate maintenance requirements. The PINN model is applied to diesel engines with a variable-geometry turbocharger and exhaust gas recirculation, using measurement data of selected state variables. The results demonstrate the ability of the PINN model to predict simultaneously both unknown parameters and dynamics accurately with both clean and noisy data, and the importance of the self-adaptive weight in the loss function for faster convergence. The input data for these simulations are derived from actual engine running conditions, while the outputs are simulated data, making this a practical case study of PINN's ability to predict real-world dynamical systems. The mean value model of the diesel engine incorporates empirical formulae to represent certain states, but these formulae may not be generalizable to other engines. To address this, the study considers the use of deep neural networks (DNNs) in addition to the PINN model. The DNNs are trained using laboratory test data and are used to model the engine-specific empirical formulae in the mean value model, allowing for a more flexible and adaptive representation of the engine's states. In other words, the mean value model uses both the PINN model and the DNNs to represent the engine's states, with the PINN providing a physics-based understanding of the engine's overall dynamics and the DNNs offering a more engine-specific and adaptive representation of the empirical formulae. By combining these two approaches, the study aims to offer a comprehensive and versatile approach to monitoring the health and performance of diesel engines.

cs.LG↗

arXiv2023

Deep neural operator for learning transient response of interpenetrating phase composites subject to dynamic loading

Additive manufacturing has been recognized as an industrial technological revolution for manufacturing, which allows fabrication of materials with complex three-dimensional (3D) structures directly from computer-aided design models. The mechanical properties of interpenetrating phase composites (IPCs), especially response to dynamic loading, highly depend on their 3D structures. In general, for each specified structural design, it could take hours or days to perform either finite element analysis (FEA) or experiments to test the mechanical response of IPCs to a given dynamic load. To accelerate the physics-based prediction of mechanical properties of IPCs for various structural designs, we employ a deep neural operator (DNO) to learn the transient response of IPCs under dynamic loading as surrogate of physics-based FEA models. We consider a 3D IPC beam formed by two metals with a ratio of Young's modulus of 2.7, wherein random blocks of constituent materials are used to demonstrate the generality and robustness of the DNO model. To obtain FEA results of IPC properties, 5,000 random time-dependent strain loads generated by a Gaussian process kennel are applied to the 3D IPC beam, and the reaction forces and stress fields inside the IPC beam under various loading are collected. Subsequently, the DNO model is trained using an incremental learning method with sequence-to-sequence training implemented in JAX, leading to a 100X speedup compared to widely used vanilla deep operator network models. After an offline training, the DNO model can act as surrogate of physics-based FEA to predict the transient mechanical response in terms of reaction force and stress distribution of the IPCs to various strain loads in one second at an accuracy of 98%. Also, the learned operator is able to provide extended prediction of the IPC beam subject to longer random strain loads at a reasonably well accuracy.

cs.LG↗

arXiv2023

Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems

Physics-informed deep learning have recently emerged as an effective tool for leveraging both observational data and available physical laws. Physics-informed neural networks (PINNs) and deep operator networks (DeepONets) are two such models. The former encodes the physical laws via the automatic differentiation, while the latter learns the hidden physics from data. Generally, the noisy and limited observational data as well as the overparameterization in neural networks (NNs) result in uncertainty in predictions from deep learning models. In [1], a Bayesian framework based on the {{Generative Adversarial Networks}} (GAN) has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets. Specifically, the proposed approach in [1] has two stages: (1) prior learning, and (2) posterior estimation. At the first stage, the GANs are employed to learn a functional prior either from a prescribed function distribution, e.g., Gaussian process, or from historical data and available physics. At the second stage, the Hamiltonian Monte Carlo (HMC) method is utilized to estimate the posterior in the latent space of GANs. However, the vanilla HMC does not support the mini-batch training, which limits its applications in problems with big data. In the present work, we propose to use the normalizing flow (NF) models in the context of variational inference, which naturally enables the minibatch training, as the alternative to HMC for posterior estimation in the latent space of GANs. A series of numerical experiments, including a nonlinear differential equation problem and a 100-dimensional Darcy problem, are conducted to demonstrate that NF with full-/mini-batch training are able to achieve similar accuracy as the ``gold rule'' HMC.

math.NA↗

arXiv2023

Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions

We consider solving the forward and inverse PDEs which have sharp solutions using physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose adaptive sampling methods (ASMs) based on the residual and the gradient of the solution. We first present a residual only based ASM algorithm denoted by ASM I. In this approach, we first train the neural network by using a small number of residual points and divide the computational domain into a certain number of sub-domains, we then add new residual points in the sub-domain which has the largest mean absolute value of the residual, and those points which have largest absolute values of the residual in this sub-domain will be added as new residual points. We further develop a second type of ASM algorithm (denoted by ASM II) based on both the residual and the gradient of the solution due to the fact that only the residual may be not able to efficiently capture the sharpness of the solution. The procedure of ASM II is almost the same as that of ASM I except that in ASM II, we add new residual points which not only have large residual but also large gradient. To demonstrate the effectiveness of the present methods, we employ both ASM I and ASM II to solve a number of PDEs, including Burger equation, compressible Euler equation, Poisson equation over an L-shape domain as well as high-dimensional Poisson equation. It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASM I or ASM II algorithm, and both methods deliver much more accurate solution than original PINNs with the same number of residual points. Moreover, the ASM II algorithm has better performance in terms of accuracy, efficiency and stability compared with the ASM I algorithm.

math.NA↗

arXiv2022

NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators

Uncertainty quantification (UQ) in machine learning is currently drawing increasing research interest, driven by the rapid deployment of deep neural networks across different fields, such as computer vision, natural language processing, and the need for reliable tools in risk-sensitive applications. Recently, various machine learning models have also been developed to tackle problems in the field of scientific computing with applications to computational science and engineering (CSE). Physics-informed neural networks and deep operator networks are two such models for solving partial differential equations and learning operator mappings, respectively. In this regard, a comprehensive study of UQ methods tailored specifically for scientific machine learning (SciML) models has been provided in [45]. Nevertheless, and despite their theoretical merit, implementations of these methods are not straightforward, especially in large-scale CSE applications, hindering their broad adoption in both research and industry settings. In this paper, we present an open-source Python library (https://github.com/Crunch-UQ4MI), termed NeuralUQ and accompanied by an educational tutorial, for employing UQ methods for SciML in a convenient and structured manner. The library, designed for both educational and research purposes, supports multiple modern UQ methods and SciML models. It is based on a succinct workflow and facilitates flexible employment and easy extensions by the users. We first present a tutorial of NeuralUQ and subsequently demonstrate its applicability and efficiency in four diverse examples, involving dynamical systems and high-dimensional parametric and time-dependent PDEs.

cs.LG↗

arXiv2022

Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems

Understanding real-world dynamical phenomena remains a challenging task. Across various scientific disciplines, machine learning has advanced as the go-to technology to analyze nonlinear dynamical systems, identify patterns in big data, and make decision around them. Neural networks are now consistently used as universal function approximators for data with underlying mechanisms that are incompletely understood or exceedingly complex. However, neural networks alone ignore the fundamental laws of physics and often fail to make plausible predictions. Here we integrate data, physics, and uncertainties by combining neural networks, physics-informed modeling, and Bayesian inference to improve the predictive potential of traditional neural network models. We embed the physical model of a damped harmonic oscillator into a fully-connected feed-forward neural network to explore a simple and illustrative model system, the outbreak dynamics of COVID-19. Our Physics-Informed Neural Networks can seamlessly integrate data and physics, robustly solve forward and inverse problems, and perform well for both interpolation and extrapolation, even for a small amount of noisy and incomplete data. At only minor additional cost, they can self-adaptively learn the weighting between data and physics. Combined with Bayesian Neural Networks, they can serve as priors in a Bayesian Inference, and provide credible intervals for uncertainty quantification. Our study reveals the inherent advantages and disadvantages of Neural Networks, Bayesian Inference, and a combination of both and provides valuable guidelines for model selection. While we have only demonstrated these approaches for the simple model problem of a seasonal endemic infectious disease, we anticipate that the underlying concepts and trends generalize to more complex disease conditions and, more broadly, to a wide variety of nonlinear dynamical systems.

cs.LG↗

arXiv2022

Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed, which we will make available as open-source library of all codes included in this framework.

cs.LG↗