How much can one learn a partial differential equation from its solution?

Abstract

In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data driven and data adaptive approach based on local regression and global consistency is proposed for stable PDE identification. Numerical experiments are provided to verify our analysis and demonstrate the performance of the proposed algorithms.

How much can one learn from a single solution of a PDE?

“…In practice, this means that if one can find a linear space a single (or multiple) sampled solution(s) is (are) close to, that space can be used to approximate all solutions. Moreover, the learned solution space can be used as a general regularization for PDE learning [4] which complements the limitations of local matching.…”

“…However, fixing t 0 and t 1 and let τ → t + 0 , the norm ν(• ; τ ) L 2 [t 0 ,t 1 ] will increase rapidly according to Theorem 3.7. For u t = −Lu, where L is a self-adjoint elliptic operator, it has been shown [4] that given any ε > 0, for any solution trajectory u(x, t) on [t 0 , t 1 ], there exists a linear subspace…”

“…Particularly, the Bernoulli random perturbation has been studied in [11] and Gaussian random perturbation is considered in [12]. For u t = −Lu, where −L is a self-adjoint elliptic operator, given any ε > 0 and any solution trajectory u(x, t) on [t 0 , t 1 ], there exists a linear subspace 53) is satisfied [4]. This means σ n = O(e −n ) which makes decay very fast with respect to υ and the computation of V is sensitive to noise.…”

“…Even if the event happens again, the environment and setup is different. In our earlier work [4], we characterized the solution data space, the dimension of the space spanned by all snapshots in time of a solution with certain tolerance, and showed how it is affected by the underlying PDE and the initial data for an evolution PDE ∂ t u(x, t) = −Lu. In particular, it was shown that the data space is small if L is strongly elliptic, which implies limited data from a single solution trajectory and the challenge for PDE learning in practice.…”

A Kernel Approach for PDE Discovery and Operator Learning

“…Such theoretical ques-tions have attracted attention very recently(He, Zhao, and Zhong 2022;He et al 2022) but current theoretical results are quite scarce.While previous methods find good approximations to P, the DE that they learn can be very difficult to solve using conventional techniques such as finite differences or finite elements. Simply put, the dictionary of features can involve highly nonlinear and stiff terms that lead to numerical instabilities and may need dedicated solvers.…”