Workshop

Abstracts and titles:

Day 1：

• Title: Neural Network Approximation Theory in Terms of  Barron and Sobolev Spaces (Jinchao Xu)

Abstract: This talk presents a unified framework connecting Barron and Sobolev spaces to analyze the approximation properties of ReLU$^k$ neural networks. It establishes both classical and new sharp approximation rates, showing that for functions in the relevant Barron space, ReLU$^k$ networks can achieve high accuracy without the curse of dimensionality. The same convergence rate is obtained in the Sobolev space $H^{(d+2k+1)/2}$ for linearized ReLU$^k$ networks, and these rates are compared with similar results achievable for classical global polynomial spaces. A key insight is that the ReLU$^k$ Barron space and the Sobolev space $H^{(d+2k+1)/2}$ exhibit comparable metric entropy and complexity.

An interesting consequence is that a piecewise linear finite element space, when defined in terms of ReLU neural networks, avoids the curse of dimensionality for sufficiently smooth functions, whereas the classical linear finite element space still suffers from it. We further introduces a bit-centric perspective, showing that parameter count alone is not a reliable measure of model complexity or approximability. Collectively, these results bridge finite element analysis and deep learning theory, offering new mathematical insights into scientific machine learning.

•  Title: ADAM: A Trustworthy All-in-one Diffusion Architecture for Multi-Physics Learning (Guang Lin)

Abstract: We present ADAM (All-in-one Diffusion Architecture for Multi-Physics), a class-conditional generative framework designed to learn heterogeneous families of partial differential equations (PDEs) within a single, unified model. By treating PDE families and their physical coefficients as conditioning variables, ADAM learns a comprehensive prior over coefficients, initial conditions, and spatiotemporal evolutions. This architecture enables the seamless execution of forward prediction and inverse inference tasks by simply adapting the conditioning and guidance strategies at inference time.

Built upon the Elucidated Diffusion Models (EDM) formulation, ADAM integrates a score-based prior with observation-guided probability-flow dynamics. This allows the model to handle both scalar and vector PDEs under full or highly sparse observations. Through extensive numerical experiments—spanning diffusion, advection, Burgers, Allen–Cahn, and Navier–Stokes systems—we demonstrate that ADAM achieves state-of-the-art accuracy in forward modeling, robust reconstruction of initial conditions, and reliable parameter recovery.

To ensure "trustworthiness" in high-stakes scientific applications, we augment the generative process with conformal calibration, providing distribution-free prediction intervals with rigorous statistical coverage guarantees. Finally, we demonstrate that ADAM can perform PDE model selection and physical law discovery from as few as two snapshots. By jointly inferring the most plausible PDE family and its associated coefficients, ADAM provides a powerful, unified probabilistic framework for data-driven physical discovery and uncertainty quantification.

• Title:  Learning operators and diffusion models over function spaces (Lu Lu)

Abstract: TBA

• Title: A Data-Driven Computational Framework for Identifiability and Nonlinear Dynamics Discovery in Complex Systems (Wenrui Hao)

Abstract: Data-driven modeling is essential for deciphering complex biological systems, yet its utility is often constrained by two fundamental hurdles: the inability to guarantee parameter identifiability and the high computational cost of learning nonlinear dynamics. This talk introduces a unified computational framework designed to overcome these challenges, bridging theoretical rigor with scalable machine learning.

The first component of the framework establishes a computational foundation for practical identifiability. By leveraging the Fisher Information Matrix and its theoretical links to coordinate identifiability, we propose an efficient method for identifiability assessment. We further introduce regularization-based strategies to manage non-identifiable parameters, thereby enhancing model reliability and facilitating robust uncertainty quantification.

To address the discovery of nonlinear dynamics, we present the Laplacian Eigenfunction-Based Neural Operator (LE-NO). This operator learning framework is specifically engineered for modeling reaction–diffusion equations. By projecting nonlinear operators onto Laplacian eigenfunctions, LE-NO achieves superior computational efficiency and generalization across varying boundary conditions, effectively bypassing the limitations of large-scale architectures and data scarcity.

Finally, we demonstrate the framework's utility in the context of Alzheimer’s disease modeling. We show that this integrated approach ensures reliable parameter inference while capturing the intricate nonlinear dynamics of disease progression, providing a critical step toward the development of high-fidelity digital twins for neurodegenerative pathology.

• Title:  Learning Operators through Coefficient mappings in fixed basis Spaces (Chuqi Chen)

Abstract: Operator learning has emerged as a powerful paradigm for approximating solution operators of partial differential equations (PDEs) and other functional mappings. Classical approaches typically adopt a pointwise-to-pointwise framework, where input functions are sampled at prescribed locations and mapped directly to solution values. We propose the Fixed-Basis Coefficient to Coefficient Operator Network (FB-C2CNet), which learns operators in the coefficient space induced by prescribed basis functions. In this framework, the input function is projected onto a fixed set of basis functions (e.g., random features or finite element bases), and the neural operator predicts the coefficients of the solution function in the same or another basis. By decoupling basis selection from network training, FB-C2CNet reduces training complexity, enables systematic analysis of how basis choice affects approximation accuracy, and clarifies what properties of coefficient spaces (such as effective rank and coefficient variations) are critical for generalization. Numerical experiments on Darcy flow, Poisson equations in regular, complex, and high-dimensional domains, and elasticity problems demonstrate that FB-C2CNet achieves high accuracy and computational efficiency, showing its strong potential for practical operator learning tasks. 

• Title: Randomized Greedy Algorithms for Neural Network Optimization in Solving Partial Differential Equations (Xiaofeng Xu)

Abstract: Neural networks have demonstrated remarkable approximation capabilities for solving partial differential equations (PDEs). However, their practical effectiveness remains limited by optimization challenges, even in the case of shallow neural networks. This leads to a significant gap between theoretical approximation rates and practical convergence orders. In this talk, we address the key optimization challenges in using ReLUᵏ shallow neural networks for numerical PDEs by developing the randomized orthogonal greedy algorithm (ROGA). We prove that the orthogonal greedy algorithm achieves the optimal convergence rate of ReLUᵏ shallow neural networks for solving PDEs that admit strongly convex and smooth energy functionals, thereby extending the applicability of OGA to a wide range of variational problems. We further tackle practical challenges in implementing OGA by proposing a randomized variant - the randomized orthogonal greedy algorithm. Comprehensive numerical experiments on a range of linear and nonlinear PDEs confirm its practical effectiveness and optimal convergence behavior. These results demonstrate that our proposed algorithm substantially narrows the gap between theory and practice, validating ROGA as a promising new tool for numerical PDEs.

• Title:  Learning Unknowns and Solutions: A Numerical and Data-Driven Approach to Nonlinear PDEs (Sun Lee)

Abstract: We present integrated numerical and data-driven strategies for nonlinear PDEs. First, we address core computational challenges: bifurcation analysis reveals stability conditions for the Allen-Cahn equation, and a novel companion-matrix multigrid method efficiently computes multiple equilibrium solutions. Second, we tackle PDEs with unknown components. A neural network infers spatially-varying diffusion coefficients from graph data, while an optimal control framework determines unknown therapeutic source terms in biomedical models. This work demonstrates the essential synergy between foundational numerical methods and modern learning-based approaches for complex scientific problems.

• Title:  ZENN: A Thermodynamics-Inspired Computational Framework for Heterogeneous Data-Driven Modeling (Shun Wang)

Abstract: The increasing availability of complex, heterogeneous datasets poses significant challenges for traditional data-driven methods, which often assume data homogeneity and fail to account for internal disparities. Quantifying entropy and its evolution in such settings remains a fundamental problem in digital twins and data science. While traditional entropy-based approaches provide useful approximations, they are limited in handling multi-source, dynamically evolving systems. To address these challenges, we introduce a zentropy-enhanced neural network (ZENN)-a novel framework that extends zentropy theory from quantum and statistical mechanics to data science by assigning intrinsic entropy to each dataset. ZENN simultaneously learns both Helmholtz energy and intrinsic entropy, enabling robust generalization, accurate high-order derivative prediction, and adaptability to heterogeneous, real-world data.

Day 2:

• Title: Accelerated Gradient Methods through Variable and Operator Splitting (Long Chen)

Abstract: We propose a Transformer-based deep direct sampling method for a class of boundary value inverse problems, enabling real-time reconstruction through a learned inverse operator mapping carefully designed boundary measurements to images. Inspired by classical direct sampling methods, one-dimensional boundary data are preprocessed via a PDE–based feature map to generate multi-frequency two-dimensional harmonic extensions as input channels. By introducing a learnable nonlocal kernel, the direct sampling approximation is recast as a modified attention mechanism within the Transformer architecture. The method is applied to electrical impedance tomography (EIT), a prototypical severely ill-posed nonlinear inverse problem, where numerical results demonstrate improved reconstruction accuracy over existing direct sampling approaches and contemporary operator-learning methods, together with strong robustness to measurement noise. This work illustrates how attention mechanisms, originally developed for natural language processing, can be systematically adapted to incorporate a priori mathematical structure, leading to neural architectures that are more compatible with the underlying physics of inverse problems.

This is a joint work with Ruchi Guo (Sichuan University) and Shuhao Cao (University of Missouri-Kansas City).

• Title: Approximative reasoning based on probabilistic natural deduction (Marija Boricic Joksimovic)

Abstract: We present two probabilistic variations NKprob and NKprob(\epsilon) on crisp Gentzen's natural deduction system NK for classical propositional logic. These probabilistic natural deduction systems can be considered a mixture of the proof--theoretic ideas of Gentzen and Prawitz and the probability concept of propositions of Carnap and Popper. The developed systems of inference rules enables us to express the fact that $C^p$ can be inferred from $A^m$ and $B^n$, i.e. $A^m,B^n\vdash C^p$, with a traditional deduction relation, where $A^m,B^n$ and $C^p$ are the probabilized sentences $A,B$ and $C$. We prove that our systems NKprob and NKprob(epsilon) are sound and complete with respect to the traditional Carnap--Popper type of semantics.

• Title: Stabilization Methods for General Convection-Diffusion Equations (Jindong Wang)

Abstract: We will present several stabilized finite element methods for advection–diffusion equations of different differential forms, with particular emphasis on recent extensions to vector-valued problems. We introduce a class of unwinding-type schemes, together with exponentially fitted methods, and also present a robust multigrid solver for H(curl) convection–diffusion problems. These developments illustrate concrete strategies for the design and analysis of discretizations for general convection–diffusion problems, and demonstrate how classical scalar techniques can be systematically generalized to accommodate the distinctive mathematical structure of vector field spaces.

• Title: Dissecting Neural Operators: A Numerical Analyst's Perspective (Shuhao Cao)

Abstract: Neural operators, which can represent operator-valued data, have emerged as powerful surrogates for partial differential equation solution mappings, offering the ability to learn solutions for parametric problems in a resolution-invariant manner. In this talk, we review popular architectures and present some results on the layer-wise approximation properties and representational capacities of neural operators. However, most of them are density/existence results. Inspired by standard practices in numerical analysis, we present several experiments to explore possible forms of regularity-based bounds.

• Title: Greedy Algorithms for Neural Networks Approximation for Definite and Indefinite Problems (Qingguo Hong)

Abstract: This talk will present a priori error analysis of the shallow neural network approximation to the solution to both the definite and indefinite elliptic problems and a cutting-edge implementation of the Orthogonal Greedy Algorithm (OGA) tailored to overcome the challenges of indefinite elliptic problems, which is a domain where conventional approaches often struggle due to the lack of coerciveness. A rigorous priori error analysis that shows the neural network’s ability to approximate the solution of indefinite problems is confirmed numerically by OGA. In particular, massive numerical implementations are conducted to justify the theory, some of which showcase the OGA’s superior performance in comparison to the traditional finite element method. This advancement illustrates the potential of neural networks enhanced by OGA to solve intricate computational problems more efficiently, thereby marking a significant leap forward in the application of machine learning techniques to mathematical problem-solving.

• Title:  Stepsize-Based Accelerated and Adaptive Gradient Descent on Riemannian Manifolds and Wasserstein Space (Jiyoung Park)

Abstract: One of the active research directions in convex optimization concerns the choice of stepsizes in gradient descent. While the constant stepsize provide strong theoretical guarantees, recent advances show that allowing varying stepsizes can either improve algorithmic performance or relax standard assumptions. In particular, varying stepsizes can lead to 'accelerated' convergence or enable 'adaptivity' to local smoothness. In this talk, we investigate whether these stepsize based techniques for acceleration and adaptivity can be extended to Riemannian optimization problems. We focus in particular on the 2-Wasserstein space, where such Riemannian extensions of varying stepsize methods become especially favorable.

• Title:  Operator Learning of Low-Frequency Spectral Inverses for Anisotropic Elliptic PDEs (Boyi Wang)

Abstract: We study an operator learning approach for the numerical solution of anisotropic elliptic partial differential equations, motivated by a Neumann-series representation of the inverse operator. Rather than learning solution maps or full inverse operators, we focus on learning a low-frequency spectral approximation of the inverse elliptic operator, restricted to a prescribed low-frequency subspace derived from the Laplacian.

The learned operator captures how anisotropy reshapes the low-frequency spectral structure of the elliptic operator and can be incorporated into numerical solvers in multiple ways, including as a preconditioner for Krylov subspace methods or as a component of a standalone iterative solver combined with classical smoothing.

Numerical experiments on two-dimensional anisotropic elliptic problems demonstrate improved convergence behavior and indicate that the learned low-frequency operator exhibits robust generalization across different coefficient configurations.

• Title: Unsupervised operator learning approach for dissipative equations via Onsager principle (Zhipeng Chang)

Abstract: Existing operator learning methods rely on supervised training with high-fidelity simulation data, introducing significant computational cost. In this work, we propose the deep Onsager operator learning (DOOL) method, a novel unsupervised framework for solving dissipative equations. Rooted in the Onsager variational principle (OVP), DOOL trains a deep operator network by directly minimizing the OVP-defined Rayleighian functional, requiring no labeled data, and then proceeds in time explicitly through conservation/change laws for the solution. Another key innovation here lies in the spatiotemporal decoupling strategy: the operator's trunk network processes spatial coordinates exclusively, thereby enhancing training efficiency, while integrated external time stepping enables temporal extrapolation. Numerical experiments on typical dissipative equations validate the effectiveness of the DOOL method, and systematic comparisons with supervised DeepONet and MIONet demonstrate its enhanced performance. Extensions are made to cover the second-order wave models with dissipation that do not directly follow OVP.