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Abstract This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function, meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter-dependent fibre problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form the loss function is given as a dual test norm of the residual; a central objective is to develop equivalent computable expressions. The first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, for example, by neural networks. Second, working with first-order SVFs we distinguish two scenarios: (i) the test space can be chosen as an $$L_{2}$$-space (such as for elliptic or parabolic problems) so that residuals can be evaluated directly as elements of $$L_{2}$$; (ii) when trial and test spaces for the fibre problems depend on the parameters (as for transport equations) we use ultra-weak formulations. In combination with discontinuous Petrov–Galerkin concepts the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter-dependent convection fields. 

Abstract Approximating functions of a large number of variables poses particular challenges often subsumed under the term “Curse of Dimensionality” (CoD). Unless the approximated function exhibits a very high level of smoothness the CoD can be avoided only by exploiting some typically hiddenstructural sparsity. In this paper we propose a general framework for new model classes of functions in high dimensions. They are based on suitable notions ofcompositional dimension-sparsityquantifying, on a continuous level, approximability by compositions with certain structural properties. In particular, this describes scenarios where deep neural networks can avoid the CoD. The relevance of these concepts is demonstrated forsolution manifoldsof parametric transport equations. For such PDEs parameter-to-solution maps do not enjoy the type of high order regularity that helps to avoid the CoD by more conventional methods in other model scenarios. Compositional sparsity is shown to serve as the key mechanism for proving that sparsity of problem data is inherited in a quantifiable way by the solution manifold. In particular, one obtains convergence rates for deep neural network realizations showing that the CoD is indeed avoided. 

Abstract The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus. 

Abstract Given integers$$n> k > 0$$ $n>k>0$ , and a set of integers$$L \subset [0, k-1]$$ $L\subset [0,k-1]$ , anL-systemis a family of sets$$\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) $$ $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ such that$$|F \cap F'| \in L$$ $|F\cap F^{\prime }|\in L$ for distinct$$F, F'\in \mathcal {F}$$ $F,F^{\prime }\in F$ .L-systems correspond to independent sets in a certain generalized Johnson graphG(n, k, L), so that the maximum size of anL-system is equivalent to finding the independence number of the graphG(n, k, L). TheLovász number$$\vartheta (G)$$ $\vartheta \left(G\right)$ is a semidefinite programming approximation of the independence number$$\alpha $$ $\alpha$ of a graphG. In this paper, we determine the leading order term of$$\vartheta (G(n, k, L))$$ $\vartheta \left(G\right(n,k,L\left)\right)$ of any generalized Johnson graph withkandLfixed and$$n\rightarrow \infty $$ $n\to \infty$ . As an application of this theorem, we give an explicit construction of a graphGonnvertices with a large gap between the Lovász number and the Shannon capacityc(G). Specifically, we prove that for any$$\epsilon > 0$$ $ϵ>0$ , for infinitely manynthere is a generalized Johnson graphGonnvertices which has ratio$$\vartheta (G)/c(G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/c\left(G\right)=\Omega \left(n^{1-ϵ}\right)$ , which improves on all known constructions. The graphGa fortiorialso has ratio$$\vartheta (G)/\alpha (G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/\alpha \left(G\right)=\Omega \left(n^{1-ϵ}\right)$ , which greatly improves on the best known explicit construction. 

Minimizing partial differential equation (PDE)-residual losses is a common strategy to promote physical consistency in neural operators. However, standard formulations often lack variational correctness, meaning that small residuals do not guarantee small solution errors due to the use of non-compliant norms or ad hoc penalty terms for boundary conditions. This work develops a variationally correct operator learning framework by constructing first-order system least- squares (FOSLS) objectives whose values are provably equivalent to the solution errors in PDE-compliant norms. We demonstrate this framework on the stationary diffusion and linear elasticity equations, incorporating mixed Dirichlet-Neumann boundary conditions via variational lifts to preserve norm equivalence without inconsistent penalties. To ensure the function space conformity required by the FOSLS loss, we propose a Reduced Basis Neural Operator (RBNO). The RBNO predicts coefficients for a pre-computed, conforming reduced basis, thereby ensuring variational stability by design while enabling efficient training. We provide a rigorous convergence analysis that bounds the total error by the sum of finite element discretization error, reduced basis projection error, neural network approximation error, statistical estimation error, and optimization error. Numerical benchmarks validate these theoretical bounds and demonstrate that the proposed approach achieves superior accuracy in PDE-compliant norms compared to standard baselines, while the residual loss serves as a reliable and computable a posteriori error estimator. 

This paper develops expansive gradient dynamics in deep neural network (DNN)-induced map- ping spaces. Specifically, we introduce a framework for minimizing a broad class of energy func- tionals in an abstract Hilbert space setting, with applications including PDE-based problems and supervised learning. Within this framework, we study a projected gradient descent method oper- ating on DNN-parameterized sets. More importantly, we develop an adaptive expansion strategy to dynamically enlarge the DNN architecture. This expansion mechanism aims to enhance the alignment of the neural manifold-induced natural gradient direction, as well as possible, with the ideal Hilbert space gradient descent direction by leveraging the fact that we can evaluate pro- jections of the Hilbert space gradient. We demonstrate the efficacy of the proposed strategy on several simple model problems involving energies arising in the context of function approxima- tion, physics informed learning, and model reduction. In addition, we highlight the importance of assembling the neural flow matrix based on the inner product of the ambient Hilbert space. The presented algorithms represent the simplest specifications of the broader framework, with a detailed analysis deferred to future work.