apply potential theory

Green's functions characterize the fundamental solutions of partial differential equations; they are essential for tasks ranging from shape analysis to physical simulation, yet they remain computationally prohibitive to evaluate on arbitrary geometric discretizations. We present Variational Green's Function (VGF), a method that learns a smooth, differentiable representation of the Green's function for linear self-adjoint PDE operators, including the Poisson, the screened Poisson, and the biharmonic equations. To resolve the sharp singularities characteristic of the Green's functions, our method decomposes the Green's function into an analytic free-space component, and a learned corrector component. Our method leverages a variational foundation to impose Neumann boundary conditions naturally, and imposes Dirichlet boundary conditions via a projective layer on the output of the neural field. The resulting Green's functions are fast to evaluate, differentiable with respect to source application, and can be conditioned on other signals parameterizing our geometry.

This study addresses the absence of formal verification for the solvability of Dirichlet problems associated with second-order linear elliptic operators. Building upon Lean 4 and Mathlib, this work independently constructs a self-contained formalization framework for Sobolev spaces. By rigorously proving the Lax–Milgram theorem, Rellich–Kondrachov compactness, and the Fredholm alternative, it achieves end-to-end machine-verified reasoning from the existence of weak solutions to regularity estimates. Adopting a complete proof strategy entirely free of `sorry` tactics, the project systematically formalizes core results including Poincaré’s inequality, the spectral theorem, and Sobolev embeddings. Consequently, this research establishes a highly trustworthy formal foundation for classical solvability theory in partial differential equations.

This work addresses the challenge of simultaneously achieving high accuracy, mesh-free flexibility, and structural fidelity to the governing equations in solving three-dimensional harmonic potential boundary-value problems. The authors propose a novel approach that leverages the Whittaker integral representation to express the solution as a holomorphic function of complex variables, which is then approximated using a holomorphic neural network. By training solely through boundary collocation—without requiring domain-based PDE residual terms—the method constructively satisfies the governing Laplace equation exactly. This is the first integration of holomorphic neural networks with harmonic potential theory, enabling highly accurate, globally controllable solutions for both scalar and vector fields in three-dimensional Laplace and linear elasticity problems, thereby significantly enhancing the accuracy and robustness of mesh-free deep learning methods.

This work addresses the limitations of traditional numerical methods—prohibitive computational cost in high-dimensional and geometrically complex settings—and the lack of theoretical guarantees and poor generalization in current machine learning approaches for solving partial differential equations (PDEs). To bridge this gap, the authors propose a hybrid PDE-solving paradigm that integrates deductive numerical schemes with inductive learning models. They establish a unified evaluation framework encompassing six core computational challenges and, from an epistemological perspective, formally distinguish between the two methodological classes. By introducing a structure inheritance mechanism and an error budget decomposition, they clarify the conditions under which theoretical guarantees propagate through the hybrid system. Leveraging physics-informed neural networks, differentiable programming, foundation models, and quantum algorithms, the study constructs a multi-paradigm collaborative framework and articulates responsible criteria for method selection, revealing three forms of complementarity that enable scalable, theoretically grounded simulation of high-dimensional complex systems.

Data-driven discovery of partial differential equations (PDEs) remains challenging due to structural ambiguity and sensitivity to noise and data scale. Method: We propose an adjoint-based parametric modeling framework for PDE discovery. A sparse candidate library—comprising linear/nonlinear terms and spatial derivatives—is used to parameterize the PDE form, yielding a PDE-constrained optimization problem. For the first time, we systematically derive the adjoint equations for general parametric PDE families via variational calculus, enabling machine-precision analytical gradient computation. Contribution/Results: Our method significantly outperforms sparse regression approaches (e.g., PDE-FIND) in structural identification accuracy and noise robustness across diverse PDEs—including Burgers, KdV, and reaction-diffusion equations—especially under high noise levels and large-scale data. Integrated forward and adjoint numerical solvers ensure efficient training, while analytical gradients substantially accelerate optimization convergence.

This study addresses the limited generalizability and repeated training requirements of solving elliptic partial differential equations on deforming domains. To overcome these limitations, this work proposes a Neural Harmonic Measure Operator that leverages geometry-dependent harmonic measures to decouple boundary data, parameterizing the boundary probability density via a Transformer kernel. By integrating Walk-on-Spheres sampling with Poisson decomposition and employing an auxiliary network to circumvent singular volume integrals, the proposed framework enables zero-shot inference for arbitrary boundary conditions and source terms after a single training phase. Comprehensive evaluations demonstrate that the method consistently outperforms four baseline models on the 3D MCB-B benchmark, while achieving 2D performance comparable to mainstream neural operators.

This study addresses the prohibitive computational cost of traditional mesh-based solvers for the $p$-Laplace equation in three-dimensional domains when $p$ is large. To this end, it proposes an efficient solution framework based on physics-informed neural networks (PINNs) and deep operator networks (DeepONets). Theoretically, a conditional convergence result for PINNs is established, and the universal approximation capability of DeepONets for parameterized $p$-Poisson problems is rigorously proved. Numerical experiments demonstrate that the proposed approach effectively overcomes the computational bottleneck associated with high-dimensional nonlinear problems, achieving significantly superior accuracy and efficiency compared to conventional solvers. Consequently, this work provides a new paradigm for complex three-dimensional nonlinear analysis.

This work addresses the absence of a rigorous formalization of the Laplace transform and its inversion in existing interactive theorem provers. It presents the first complete formalization in Lean 4 of the Laplace transform for complex-valued functions, along with fundamental operational rules and a Bromwich-type inversion theorem grounded in real integrals and Dirichlet integrals. By integrating classical analysis with formal verification techniques, the framework is successfully applied to the harmonic oscillator problem. The development not only verifies core aspects of Laplace transform theory but also formally derives the solution and proves that its transform coincides with the standard transform of $\sin(\omega t)$. This demonstrates the feasibility and rigor of formalized mathematics in engineering analysis.

Current evaluation practices for partial differential equation (PDE) discovery lack a unified standard, and existing metrics are often narrow in scope, failing to simultaneously account for predictive accuracy, physical consistency, interpretability, and out-of-distribution generalization—potentially leading to erroneous identification of novel physical laws. This work establishes the first systematic classification framework for post-discovery PDE evaluation, integrating techniques from machine learning, numerical analysis, information theory, and symbolic regression to holistically assess model performance across multiple dimensions, including prediction fidelity, adherence to physical constraints, model simplicity, and generalization capability. By exposing the limitations of prevailing evaluation approaches, this study proposes a standardized and extensible evaluation paradigm that provides both algorithm developers and scientific practitioners with a rigorous methodological foundation for reliably validating newly discovered physical laws.

This work addresses the challenge of achieving high-order Sobolev accuracy in physics-informed neural networks (PINNs) for elliptic Dirichlet boundary value problems. The authors propose boundary-adaptive PINNs that exactly embed Dirichlet conditions into the network output by multiplying with a first-order normalized smooth boundary distance function ρ. Through a combined analysis grounded in approximation theory and statistical learning, they establish—for the first time—that merely satisfying boundary conditions is insufficient to guarantee an $H^2(\Omega)$ error bound, and rigorously identify the necessity of the first-order normalization property for ρ. Leveraging ReQU or tanh activation functions, they derive novel VC-dimension bounds for derivative hypothesis spaces and high-order Sobolev approximation rates for shallow networks, leading to a provable $H^2$ priori error estimate. Numerical experiments confirm that a properly constructed ρ significantly enhances both accuracy and convergence in MET computations.