Overview

Typically, differential equations are solved numerically or analytically based on a series of assumptions of ideal conditions. However, the variables and data used to make those assumptions can be incorrect or noisy, which results in inaccurate solutions. We have demonstrated a way to use a probability distribution to solve those differential equations even when working with a messy data set.

Market Opportunity

In recent years, the world has witnessed the emergence of a new data-driven era in which probability and statistics have been the focal point in the development of disruptive technologies such as probabilistic machine learning. This wave of change is steadily making its way into applied mathematics, giving rise to new, probabilistic interpretations of classical deterministic scientific methods and algorithms.

Karniadakis’s team’s innovative approach points the way to a future of computing with probability distributions rather than solely relying on deterministic thinking. This line of research has inspired a resurgence in probabilistic methods and algorithms that offer a robust handling of uncertainty. Such developments are defining a new area of scientific research in which probabilistic machine learning and classical scientific computing coexist in unison.

This method can be useful for using differential equations to model the properties of a building, bridge, fluid, or any other object. Those properties could include movement, displacement, variation, or any change in sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, quantum mechanics, or heat transfer.

Innovation and Meaningful Advantages

For more than two centuries, the standard way to obtain solutions to differential equations has been an analytical or numerical approach based on typically well-behaved forcing and boundary conditions for well-posed problems. Karniadakis and colleagues are changing this paradigm in a fundamental way, as they have established an interface between probabilistic machine learning and differential equations.

Karniadakis and colleagues developed data-driven algorithms for general linear equations using Gaussian process priors tailored to the corresponding integro-differential operators. The proposed algorithms can learn from scattered noisy data of variable fidelity and return solution fields with quantified uncertainty. This general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time integration and is scalable to high dimensions.

Collaboration Opportunity

We are seeking licensing opportunities to further develop this innovative technology.

Principal Investigator

IP Information

US Patent 10,963,540, Issued March 30, 2021

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