Reduced-order modeling for parameterized PDEs via implicit neural representations

We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR), which models physics signals in a continuous manner and independent of spatial/temporal discretization. The proposed framework encodes PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics characterized by multiple PDE parameters. PNODE can be inferred by a hypernetwork to reduce the potential difficulties in learning PNODE due to a complex multilayer perceptron (MLP). The framework uses an INR to decode the latent dynamics and reconstruct accurate PDE solutions. Further, a physics-informed loss is also introduced to correct the prediction of unseen parameter instances. Incorporating the physics-informed loss also enables the model to be fine-tuned in an unsupervised manner on unseen PDE parameters. A numerical experiment is performed on a two-dimensional Burgers equation with a large variation of PDE parameters. We evaluate the proposed method at a large Reynolds number and obtain up to speedup of O(10^3) and ~1% relative error to the ground truth values.Comment: 9 pages, 5 figures, Machine Learning and the Physical Sciences Workshop, NeurIPS 202

A Generalized Endogenous Grid Method for Default Risk Models

Default risk models have been widely employed to assess the ability of households and sovereigns to insure themselves against shocks. Grid search has often been used to solve these models because the complexity of the problem prevents the use of faster but less general methods. In this paper, we propose an extension of the endogenous grid method for default risk models, which is faster and more accurate than grid search. In particular, we find that our solution method leads to a more accurate bond price function, thus making substantial differences in the model’s main predictions. When applied to Arellano’s (2008) model, our approach predicts a standard deviation of the interest rate spread one-third lower and defaults 3 to 5 times less frequently than does the conventional approach. On top of that, our method is efficient. It is approximately 4 to 7 times faster than grid search when applied to a canonical model of Arellano (2008) and 19 to 27 times faster than grid search when applied to the richer model of Nakajima and R´ıos-Rull (2014). Finally, we show that our method is applicable to a broad class of default risk models by characterizing sufficient conditions

A Generalized Endogenous Grid Method for Models with the Option to Default

We develop an endogenous grid method for models with the option to default in which price schedules are endogenously determined in equilibrium and depend on individuals’ states. The algorithm has noticeable computational benefits in efficiency and accuracy. We obtain these computational benefits by combining Fella’s (2014) identification for non-concave regions with our algorithm that numerically searches for risky borrowing limits. These two procedures identify the region of solution sets to which Carroll’s (2006) endogenous grid method is applicable. To demonstrate the method, we apply our method to Nakajima and Rios-Rull’s(2014) model. In terms of computation time, this method is seven to twenty-seven times faster than the conventional grid search method. Moreover, various types of accuracy tests indicate that our method yields more accurate results than the grid search method

Effects of Temperature on Development and Voltinism of Chaetodactylus krombeini (Acari: Chaetodactylidae): Implications for Climate Change Impacts

Temperature plays an important role in the growth and development of arthropods, and thus the current trend of climate change will alter their biology and species distribution. We used Chaetodactylus krombeini (Acari: Chaetodactylidae), a cleptoparasitic mite associated with Osmia bees (Hymenoptera: Megachilidae), as a model organism to investigate how temperature affects the development and voltinism of C. krombeini in the eastern United States. The effects of temperature on the stage-specific development of C. krombeini were determined at seven constant temperatures (16.1, 20.2, 24.1, 27.5, 30.0, 32.4 and 37.8°C). Parameters for stage-specific development, such as threshold temperatures and thermal constant, were determined by using empirical models. Results of this study showed that C. krombeini eggs developed successfully to adult at all temperatures tested except 37.8°C. The nonlinear and linear empirical models were applied to describe quantitatively the relationship between temperature and development of each C. krombeini stage. The nonlinear Lactin model estimated optimal temperatures as 31.4, 32.9, 32.6 and 32.5°C for egg, larva, nymph, and egg to adult, respectively. In the linear model, the lower threshold temperatures were estimated to be 9.9, 14.7, 13.0 and 12.4°C for egg, larva, nymph, and egg to adult, respectively. The thermal constant for each stage completion were 61.5, 28.1, 64.8 and 171.1 degree days for egg, larva, nymph, and egg to adult, respectively. Under the future climate scenarios, the number of generations (i.e., voltinism) would increase more likely by 1.5 to 2.0 times by the year of 2100 according to simulation. The findings herein firstly provided comprehensive data on thermal development of C. krombeini and implications for the management of C. krombeini populations under global warming were discussed

Micromechanics-Based Homogenization of the Effective Physical Properties of Composites With an Anisotropic Matrix and Interfacial Imperfections

Micromechanics-based homogenization has been employed extensively to predict the effective properties of technologically important composites. In this review article, we address its application to various physical phenomena, including elasticity, thermal and electrical conduction, electric, and magnetic polarization, as well as multi-physics phenomena governed by coupled equations such as piezoelectricity and thermoelectricity. Especially, for this special issue, we introduce several research works published recently from our research group that consider the anisotropy of the matrix and interfacial imperfections in obtaining various effective physical properties. We begin with a brief review of the concept of the Eshelby tensor with regard to the elasticity and mean-field homogenization of the effective stiffness tensor of a composite with a perfect interface between the matrix and inclusions. We then discuss the extension of the theory in two aspects. First, we discuss the mathematical analogy among steady-state equations describing the aforementioned physical phenomena and explain how the Eshelby tensor can be used to obtain various effective properties. Afterwards, we describe how the anisotropy of the matrix and interfacial imperfections, which exist in actual composites, can be accounted for. In the last section, we provide a summary and outlook considering future challenges