Data-Driven, ML-assisted Approaches to Problem Well-Posedness

TL;DR

The paper addresses how to infer well-posedness of differential equation problems from data, challenging classical a priori assumptions by using a data-driven framework that combines Physics-Informed Neural Networks (PINNs) with manifold learning and randomized linear algebra. It demonstrates, across both ODEs and PDEs, that sampling many randomized solves and analyzing the resulting ensembles with PCA reveals the effective dimensionality of the solution space and the local richness of information provided by data patches. In the ODE harmonic oscillator $x''(t)=-x(t)$, the number and placement of constraints determine whether the solution space is 1D, 2D, or empty, while in the wave equation the discretized system $A\mathbf{u}=\mathbf{b}$ shows well-posed regions with low ensemble variance and underconstrained regions with higher variance; the KS equation further shows how nonlinear PDEs exhibit localized underdetermination in data-poor regions. For overconstrained cases, the authors apply randomized iterative methods (e.g., QuantileSCRK) to drop inconsistent equations and recover meaningful, potentially weak solutions. Overall, the work provides a practical, data-driven pipeline to diagnose, quantify, and potentially rectify well-posedness issues in differential equations, bridging ML-based solvers with mathematical analysis.

Abstract

Classically, to solve differential equation problems, it is necessary to specify sufficient initial and/or boundary conditions so as to allow the existence of a unique solution. Well-posedness of differential equation problems thus involves studying the existence and uniqueness of solutions, and their dependence to such pre-specified conditions. However, in part due to mathematical necessity, these conditions are usually specified "to arbitrary precision" only on (appropriate portions of) the boundary of the space-time domain. This does not mirror how data acquisition is performed in realistic situations, where one may observe entire "patches" of solution data at arbitrary space-time locations; alternatively one might have access to more than one solutions stemming from the same differential operator. In our short work, we demonstrate how standard tools from machine and manifold learning can be used to infer, in a data driven manner, certain well-posedness features of differential equation problems, for initial/boundary condition combinations under which rigorous existence/uniqueness theorems are not known. Our study naturally combines a data assimilation perspective with an operator-learning one.

Figures (9)

• Figure 1: Sampled solutions to IVP-2 with $a=2$ (left) and IVP-3 (right). Each figure depicts a set of 150 solutions, each given by separate runs of an underconstrained PINN network. Coloring represents the derivative $\hat{x}'(0)$ of the neural network with respect to time.
• Figure 2: Left: Statistics of repeated iterative least squares minimzation solutions for (leftmost) the non-reflecting- and (second from left) the partially reflecting-BC wave equation problem. A few characteristics are included to help discuss the results. The figures also show the locations of two sets of squares (the pink and the blue ones) whose study will help estimate the solution "richness" in the corresponding parts of the domain. Right: (Colors of fitted lines match colors of patches in the corresponding left figures). The lines result from the solution PCA scaling with the perimeter of successive larger patches (see section \ref{['sec:PDEperimeterScaling']}); the relative slopes of these lines help quantify different levels of solution "richness" prescribed by the data. The order of the latter two subfigures correspond to that of the former two: first the "no reflector BC" case, and then the "partial reflector BC" case.
• Figure 3: Truth, unique SVD solution, a single representative minimization solution, the mean of several iterative solutions, the error of this mean, and pointwise variation across the ensemble of solutions. Both non-reflecting-BC (top) and reflecting-BC (bottom) wave equation problems. The data are prescribed in the orane-bounded parallelogram.
• Figure 4: Statistics of some randomized PINN solutions of the space-time T-shaped domain for KSE for the KSE problem. From left to right: the true solution, three samples of randomized solutions, the mean of the randomized solution set, the error of the mean of the set and its variance (with the two-column domain where the data are prescribed outlined in orange). The complement of the supervisory domain is the T-shaped space time solution domain.
• Figure 5: Left: heatmap of $\mathbf{A}$ with a tri-diagonal top part --corresponding to the ODE law ($y"(t)=0$),-- and indicator rows in the last $3$ positions, corresponding to (contradictory) boundary conditions; Left-middle: Multiplying the last $3$ equations in $\vb{A} \vb{x} = \vb{b}$ by a large factor forces the least squares solutions to this system to violate the ODE law more and approximate the prescribed conditions better; Right-middle: "leaving out" one of the last $3$ equations leads to a consistent solution with only $2$ boundary conditions; Right: "leaving out" one of the equations from the top operator block of $\mathbf{A}$ leads to discontinuities: solutions satisfying the equation in portions only of the domain (related to weak solutions).