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1. Representation of linear PDEs with spatial integral terms as Partial Integral Equations

   https://doi.org/10.23919/ACC55779.2023.10156465  

   Shivakumar, Sachin; Das, Amritam; Peet, Matthew M. (May 2023, Proceedings of the American Control Conference)

   In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods. 

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2. Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

   https://doi.org/10.1016/j.ifacol.2023.10.538  

   Braghini, Danilo; Peet, Matthew M. (January 2023, IFAC-PapersOnLine)

   Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables where the disturbance does not enter through the boundary. For such systems, we define a notion of H2-norm using an initial state-to-output framework and show that this notion reduces to more traditional concepts under the assumption of existence of a strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE representation and for such systems show that computing a minimal upper bound on the H2-norm of delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation results are available in the literature. The numerical results demonstrate the accuracy of the computed upper bound on the H2-norm. 

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3. Integral Quadratic Constraints with Infinite-Dimensional Channels

   https://doi.org/10.23919/ACC55779.2023.10156335  

   Talitckii, Aleksandr; Seiler, Peter; Peet, Matthew M. (May 2023, Proceedings of the American Control Conference)

   Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such inputoutput analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinitedimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinitedimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity. 

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4. State of the Art in Grid-Free Monte Carlo Methods for Partial Differential Equations

   https://doi.org/10.1145/3721241.3734001  

   Sawhney, Rohan; Miller, Bailey; Gkioulekas, Ioannis; Crane, Keenan; Jarosz, Wojciech; Zhao, Shuang; Nabizadeh, Mohammad Sina; Li, Zilu (August 2025, ACM)

   This 3 hour course provides a detailed overview of grid-free Monte Carlo methods for solving partial differential equations (PDEs) based on the walk on spheres (WoS) algorithm, with a special emphasis on problems with high geometric complexity. PDEs are a basic building block of models and algorithms used throughout science, engineering and visual computing. Yet despite decades of research, conventional PDE solvers struggle to capture the immense geometric complexity of the natural world. A perennial challenge is spatial discretization: traditionally, one must partition the domain into a high-quality volumetric mesh—a process that can be brittle, memory intensive, and difficult to parallelize. WoS makes a radical departure from this approach, by reformulating the problem in terms of recursive integral equations that can be estimated using the Monte Carlo method, eliminating the need for spatial discretization. Since these equations strongly resemble those found in light transport theory, one can leverage deep knowledge from Monte Carlo rendering to develop new PDE solvers that share many of its advantages: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations. The course is divided into two parts. Part I will cover the basics of using WoS to solve fundamental PDEs like the Poisson equation. Topics include formulating the solution as an integral equation, generating samples via recursive random walks, and employing accelerated distance and ray intersection queries to efficiently handle complex geometries. Participants will also gain experience setting up demo applications involving data interpolation, heat transfer, and geometric optimization using the open-source “Zombie” library, which implements various grid-free Monte Carlo PDE solvers. Part II will feature a mini-panel of academic and industry contributors covering advanced topics including variance reduction, differentiable and multi-physics simulation, and applications in industrial design and robust geometry processing. 

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5. Learning partial differential equations for biological transport models from noisy spatio-temporal data

   https://doi.org/10.1098/rspa.2019.0800  

   Lagergren, John H.; Nardini, John T.; Michael Lavigne, G.; Rutter, Erica M.; Flores, Kevin B. (February 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We investigate methods for learning partial differential equation (PDE) models from spatio-temporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection–diffusion, classical Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) and nonlinear Fisher–KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model. 

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