Deep Hidden Physics Models

Deep Learning of Nonlinear Partial Differential Equations

A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical models of the physical world. In the current era of abundance of data and advanced machine learning capabilities, the natural question arises as how can we automatically uncover the underlying laws of physics from high-dimensional data generated from experiments? In this work, we put forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time. Specifically, we approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks. The first network acts as a prior on the unknown solution and essentially enables us to avoid numerical differentiations which are inherently ill-conditioned and unstable. The second network represents the nonlinear dynamics and helps us distill the mechanisms that govern the evolution of a given spatiotemporal data-set. We test the effectiveness of our approach for several benchmark problems spanning a number of scientific domains and demonstrate how the proposed framework can help us accurately learn the underlying dynamics and forecast future states of the system. In particular, we study the Burgers', Korteweg-de Vries (KdV), Kuramoto-Sivashinsky, nonlinear Shrödinger, and Navier-Stokes equations.

Conic Economics

Conic Economics is an attempt to model modern general equilibria under uncertainty based on the recognition that all risks cannot be eliminated, perfect hedging is not possible, and some risk exposures must be tolerated. Therefore, we need to define the set of acceptable risks as a primitive of the financial economy. This set will be a cone, hence the word conic. Such a conic perspective challenges classical economics by introducing finance into the economic models and enables us to rewrite major chapters of classical micro- and macro-economics textbooks. The classical models dictate that economic players are able to trade the whole of their endowments at what is known as a market-clearing price and direct all proceeds to the consumption of goods and services. According to these models, the aggregate consumption does not exceed the total endowment, suggesting that finance is not a necessary component in the economy. Conic Economics proposes a case in which some gap occurs between the aggregate supply and demand whereby the financial primitives cover the aforementioned gap. This also generates a bid-ask spread at equilibrium depending on the cone of acceptable risks. This work questions the traditional law of one price and poses a direct challenge to Adam Smith's invisible hand theory. Since the housing crisis in 2008, economists and statisticians have questioned the law of one price. The implications of this academic debate are sweeping and affect players at all levels of the economy.