A new software tool aims to automate the analysis of nonlinear partial differential equations (PDEs)—the mathematical rules that govern everything from airflow over a wing to plasma inside a fusion reactor—without requiring a specialist to manually simplify them first. Engineers and scientists routinely rely on PDEs to design aircraft, synthesise chemicals, and control combustion. But nonlinear PDEs, which describe systems where small changes can produce large, unexpected effects, are notoriously difficult to analyse. Current methods force users to "lump" the continuous state into discrete points, a process that introduces errors, misses shock waves and boundary effects, and demands deep mathematical expertise. This project combines two existing mathematical frameworks—Sum-of-Squares (SOS) polynomial optimisation and the Partial Integral Equation (PIE) approach—to create a new algebra that works directly with the continuous state. The resulting software, PIESOS, will let a user declare a nonlinear PDE and receive a provable certificate of stability or energy gain, without needing to discretise the system or write complex code. If successful, the tool could accelerate the design of fusion reactors by enabling reliable analysis of plasma confinement models, and improve the safety and efficiency of fluid-flow systems in power plants and aircraft. The work is fundamental mathematics, but it directly targets a bottleneck that currently limits practical use of nonlinear PDEs in engineering.
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Partial Differential Equations (PDEs) describe the locally averaged mass interaction of large numbers of constituent elements – be they molecules, cells, organisms, or devices. Controlling such processes allows us to create lift over the wing of an aircraft, synthesize new chemicals and materials, and regulate combustion in power plants. The design of controllers for such processes, however, is difficult due to nonlinear interactions and a spatially distributed state. Specifically, our reliance on digital computers has traditionally required lumping of this distributed state into a fixed number of parameters representing discrete points in the state or weighted averages of the state. Lumping allows us to use sophisticated algorithms for the analysis and control of Ordinary Differential Equations (ODE) such as the Sum-of-Squares (SOS) framework for polynomial optimization. However, these methods require sophisticated mathematical analysis, limiting their use in speculative and data-based PDE models. Furthermore, lumping fails to capture discrete-continuum effects such as shock, fluid-structure interaction, and boundary inputs. As a result current algorithms for the analysis of nonlinear PDEs suffer from difficult implementation, low reliability, high complexity, and questionable physical interpretation. Recently, new methods have emerged for the analysis and control of linear PDEs which do not discretize the state but instead parameterize operators which act on this state – The Partial Integral Equation (PIE) framework. This approach has worked well for linear PDEs where we can parameterize linear operators using multipliers and kernels – resulting in efficient universal software tools for rapid analysis, control and simulation. For nonlinear PDEs, however, there is no obvious parametrization of nonlinear operators on a distributed state. The goal of this project, then, is to develop efficient convex algorithms for analysis of nonlinear PDEs by combining the SOS and PIE frameworks. We then demonstrate their accuracy and reliability using models of fluid flow and demonstrate their use for speculative and data-based PDEs using plasma confinement in Tokamak nuclear fusion reactors. The main contribution of this project is to propose a new multiplication algebra of polynomials on a distributed state and to parameterize that algebra using linear operators. We then adopt the fundamental state transformation used in the PIE framework to strip out partial derivatives and boundary conditions – representing the dynamics of the PDE using distributed polynomials (a nonlinear PIE). Next, we generalize the SOS approach to parametrization of positive polynomials by using positive linear operators to parameterize positive distributed polynomials. Global stability and input energy gain can then be tested by searching for polynomial Lyapunov functions whose derivative is likewise a distributed polynomial. The results are extended to local stability by generalizing positivstellensatz results from semialgebraic geometry. To obviate the need for sophisticated mathematical and computational expertise in nonlinear PDEs, we will develop a universal software tool PIESOS (modelled after the SOSTOOLS and PIETOOLS software packages) for: conversion of nonlinear PDEs to nonlinear PIEs; manipulation and optimization of positive distributed polynomials; and stability/energy gain analysis of nonlinear PDEs. PIESOS will have a user-friendly interface for declaring nonlinear PDEs and return provable certificates of stability and input energy gain. Furthermore, this software will use efficient data structures, sparsity, and Newton polytope methods to reduce computational complexity well below that needed by standard SOS-based lumping methods. Finally, the accuracy and reliability of the results will be demonstrated by application to nonlinear PDE fluid flow models.
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