It is concluded that remaining instabilities are attributable to numerical issues, providing a unifying validity foundation for operator-informed kernel methods.
Abstract
Many modern machine learning models can be understood as kernel-based function-space models, including Gaussian processes and neural tangent kernels. In scientific machine learning, differential operators are increasingly used to encode physical structure directly into such models. However, it has remained unclear under which conditions these constructions are valid machine learning models, i.e. preserve positive semi-definiteness, and whether observed instabilities arise from ill-posed modeling or numerical effects. Here, we establish a simple and sufficient condition for positive semi-definiteness: for linear differential operators of order m, the base kernel must be m-times continuously differentiable. Crucially, this guarantee holds for operators with non-constant and even discontinuous coefficients. Examples are ubiquitous in physical systems, including diffusion, material elasticity, wave propagation in inhomogeneous media, and quantum systems. We conclude that remaining instabilities are attributable to numerical issues, providing a unifying validity foundation for operator-informed kernel methods.
The universal consistency of PIKS is established for linear differential constraints, proving that for universal kernels (such as Gaussian or Mat\'ern), the estimator asymptotically learns the target while satisfying physical constraints.
Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria et al.· arXiv.org· 1 citation
This work introduces a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction, and develops this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces.
Adrien Weihs, Chun-Yang Liao, Jingmin Sun et al.· 0 citations
This work extends the orthogonal polynomial kernel paradigm by introducing a novel family based on discrete Hermite I polynomials, a class of $q$-orthogonal polynomials that generalize classical Hermite polynomials through a deformation parameter $q$.
Álvaro Sánchez-Paniagua Ríos, J. P. Llerena, Alberto Lastra et al.· 0 citations
Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.
Gnankan Landry Regis N'guessan, Bum Jun Kim· 0 citations
Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.
Bai-Ming Zhang, Jin-Song Tang, Ying Xu et al.· 0 citations
Flows on measure spaces have long been examined in stochastic analysis and have recently attracted significant interest in machine learning, leading to intriguing research questions that often fall outside the scope of existing theory. Normalizing flows, score-based diffusion, and flow matching models are among the most powerful generative neural methods and rely on the geometry of measure spaces. In particular, the Wasserstein metric and optimal transport techniques have advanced the field in recent years. However, involving different Riemannian-like metrics on measure spaces, e.g., by the framework of right-invariant metrics on the group of diffeomorphisms and their action on objects, e.g., densities, and designing transport inference functionals with advanced properties like equivariance led to new neural models. Generative models can be conditioned on (degraded) data, which leads to new developments in the solution of Bayesian inverse problems. Viewing transformers as interacting particle systems introduced a new mathematical perspective on these complex systems and shed light on their clustering behavior. Finally, learning neural models comes with new challenges in (stochastic) optimization, such as accelerated optimization, operator splitting, and mirror descent on measure spaces, ensemble filtering methods, the treatment of high dimensions via slicing or Fourier random features, as well as scalability questions and related lifting to infinite-dimensional spaces. The workshop will bring together scientists interested in different aspects of flows on measure spaces to further understand and develop their analysis, in particular to address questions in deep generative learning and to develop improved optimization methods for measure spaces.
P. Rigollet, Giuseppe Savaré, Gabriele Steidl et al.· Oberwolfach Reports· 0 citations
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