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版本更新 · v3arXiv:2604.18540

Duality for the Adversarial Total Variation

Leon Bungert, Lucas Schmitt

原文摘要

Adversarial training of binary classifiers can be reformulated as regularized risk minimization involving a nonlocal total variation. Building on this perspective, we establish a characterization of the subdifferential of this total variation using duality techniques. To achieve this, we derive a dual representation of the nonlocal total variation and a related integration-by-parts formula, involving a nonlocal gradient and divergence. We provide such duality statements both in the space of continuous functions vanishing at infinity on proper metric spaces and for the space of essentially bounded functions on separable metric spaces. Furthermore, under some additional conditions we provide characterizations of the subdifferential in these settings.

math.AP · cs.LG · math.FA · math.OC

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.14468

异质SIS反应扩散模型的贝叶斯反问题与流行病阈值不确定性

Bayesian inverse problems and epidemic-threshold uncertainty for a heterogeneous SIS reaction-diffusion model

Duc-Lam Duong

AI 速读 基于摘要

该文研究在异质SIS反应扩散模型中，如何利用早期感染观测对疾病持续阈值进行贝叶斯推断，并给出低流行极限下的后验收敛与阈值概率误差界。

math.AP · cs.LG · math.NT · math.ST · q-bio.PE

arXiv ↗PDF ↗
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We study Bayesian inference of persistence from early observations of a heterogeneous susceptible--infected--susceptible (SIS) reaction--diffusion model with standard incidence. We show that the leading prevalence dynamics depend on the unknown coefficients only through the difference between the transmission and recovery rates and the infected diffusivity. The same pair determines the principal eigenvalue whose sign governs persistence, so early prevalence can inform the threshold even when the individual rates remain uncertain. We prove global weak well-posedness and quantitative parameter stability for the SIS system and use the resulting observation-map estimates to establish Bayesian well-posedness. For an initial infection of size $\varepsilon$, we derive a uniform second-order expansion that describes how the separate rates enter prevalence. Incidence depends explicitly on transmission at leading order. When the noise standard deviation is proportional to $\varepsilon$, we prove $O(\varepsilon)$ convergence in Hellinger distance of the nonlinear posteriors to their low-prevalence limits for both prevalence alone and joint prevalence and incidence data. We bound the principal eigenvalue using the growth rate of the infected $L^2$ norm between two times. This rate gives a lower bound for the full nonlinear system, and the upper bound separates spatial transients from nonlinear saturation. Uniform spectral estimates and bounds on infection size and measurement error give sufficient conditions for persistence and extinction. We also bound the error in posterior persistence probabilities under forward and eigenvalue approximation, accounting for posterior mass near the threshold. Numerical examples illustrate threshold inference and the predicted \(O(\varepsilon)\) convergence of the nonlinear posteriors to their low-prevalence limits in Hellinger distance.

版本更新 · v3arXiv:2604.23599

Partition-of-Unity Gaussian Kolmogorov-Arnold Networks

Amir Noorizadegan

原文摘要

Gaussian basis functions provide an efficient and flexible alternative to spline activations in Kolmogorov-Arnold networks (KANs), but their performance is sensitive to the Gaussian scale parameter ε. We introduce the partition-of-unity Gaussian KAN (PU-GKAN), a kernel-based normalization framework that transfers the Shepard partition-of-unity principle to trainable KAN edge functions. Each Gaussian edge basis is normalized by its local sum over fixed centers, preserving the standard edge-based architecture and essentially the same trainable coefficient structure as GKAN. The resulting feature map reproduces constants exactly at the edge level and admits an explicit normalized finite-feature kernel interpretation. Numerical experiments show that PU-GKAN reduces sensitivity to ε, improves validation accuracy for most smooth and moderately non-smooth targets, and provides more stable training behavior. These benefits persist across variations in sample size and number of centers, higher-dimensional architectures, and Matern RBF bases. PU-GKAN also improves physics-informed neural networks for Helmholtz and wave equations, with validation-error reductions of up to 90%. These results establish partition-of-unity normalization as a mathematically grounded and broadly applicable framework for improving the reliability and accuracy of radial basis function KANs.

cs.CE · cs.AI · math.AP

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.10570

Deep backward multistep high-order regression methods for high dimensional nonlinear parabolic partial differential equations

Qiang Han, Shaolin Ji, Zihao Xv

原文摘要

We propose a deep backward multistep high-order regression (MHDBR) method for semilinear parabolic partial differential equations through their backward stochastic differential equation representations. The method combines multistep time discretization with regression targets formed by conditional expectations. Under explicit regularity, starting-value, forward-approximation and network-moment assumptions, we prove an error bound separating time discretization, conditional Monte Carlo sampling and neural-network approximation. The analysis assumes exact population minimization; finite-training-sample generalization and optimization errors are not included. For generators independent of the gradient variable, suitably constrained GroupSort networks and sufficiently large conditional sample sizes give order $J$ in the stated nodal $L^2$ error norm. The required network sizes depend on dimension, so this result does not establish dimension-independent complexity. We present a reproducible time-discretization test satisfying the analytical assumptions and three broader neural-network benchmarks, whose relation to the assumptions is discussed explicitly.

stat.ML · cs.LG · math.AP

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2605.19823

Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions

Ha Dang, Sebastian Schmidt, Juergen Hesser

原文摘要

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.

cs.LG · cs.AI · math.AP · math.DS · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2604.18261

DeepRitzSplit: A Variational Neural Operator Framework for Gradient Flows with Application to Dendritic Growth

Benoît Appolaire, Ludovick Gagnon, Chih-Kang Huang, Miha Založnik

原文摘要

Phase-field models are widely used to simulate microstructure evolution but their numerical solution remains computationally demanding. Neural operator surrogates provide a promising alternative, yet existing approaches are data-driven, requiring large datasets and exhibiting limited robustness when extrapolating to unseen configurations. We introduce DeepRitzSplit, an unsupervised neural operator framework for gradient-flow systems based on a variational formulation of energy-stable splitting schemes. It learns the time evolution semigroup associated with a convex-concave free-energy decomposition by minimizing the underlying variational problem at each time step. It is designed to inherit the dissipative structure of the underlying gradient flow without requiring trajectory datasets. We further introduce a Reaction-Diffusion Neural Operator (RDNO) architecture motivated by splitting schemes, with separate reaction and diffusion components to improve training efficiency and interpretability. We assess the approach on the Allen-Cahn equation and apply it to a phase-field model of anisotropic dendritic growth during solidification. In the Allen-Cahn benchmark, DeepRitzSplit shows better generalization than the supervised baseline on the out-of-distribution configurations tested, using 180 training configurations. In dendritic growth, the method captures the overall phase morphology and dendrite tip velocity well, while the tip radius is more sensitive to the training strategy. Refinement based on the residual of the splitting scheme markedly improves this second-order morphological quantity. The trained operators retain good predictive capabilities on configurations substantially different from the training set. These results demonstrate the potential of variationally trained neural operators as physically informed surrogates for accelerated microstructure simulations.

math.AP · cs.LG · math.NA

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.07395

Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision

Michal Shavit, Yongji Wang, Tristan Buckmaster

原文摘要

We present a computational framework that combines machine-learning-based discovery with rigorous computer-assisted numerical certification and formal verification for strongly nonlinear partial differential equations. We demonstrate the approach for periodic gravity-capillary traveling water waves. Physics-informed neural networks are used as a search mechanism, with the wave profile, wave speed and, in continuation experiments, surface tension learned directly from the governing equations without training data. Through a staged training and refinement procedure, the neural networks achieve near-machine-precision residuals of the governing equations, even in strongly nonlinear regimes. A single formulation recovers high-amplitude Stokes waves, Crapper waves and Wilton ripples. Continuation from a high-amplitude Wilton ripple in surface tension reveals new high-amplitude candidate solutions. We freeze one candidate as an explicit finite representation and rigorously certify a uniform bound on its governing-equation residual using interval arithmetic, accounting for floating-point roundoff, Fourier truncation, aliasing and grid-to-continuum errors. This independently validates the neural-network accuracy. We formalize the certification argument in Lean, which machine-checks the deductions from certified numerical bounds to the residual theorem. Both the interval certification and formalization were developed with substantial assistance from a large language model, while the validity of the result remains independent of the language model: the numerical bounds are rigorously enclosed by interval arithmetic and the subsequent mathematical deductions are checked by the Lean kernel. Together, these steps provide a pipeline from neural discovery to rigorous numerical certification and machine-checked verification with auditable mathematical guarantees.

math.AP · cs.LG · physics.flu-dyn

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06954

Certified folding-free deformable image registration: A whole-cell Jacobian certificate and a sequential variational solver

Baihan Zhao, Zicong Zhou, Guojun Liao

原文摘要

Registration maps propagate labels, measure local volume change, and map results back through their inverses; requiring invertibility. Voxelwise checks can miss folds between grid points. We study the canonical continuous extension of a nodal displacement field by trilinear interpolation and give a test that proves its Jacobian determinant positive in every grid cell or rejects the map. The test is rigorous under IEEE 754 binary64 arithmetic (correct rounding, no contraction). With a fixed boundary, certification implies a bi-Lipschitz homeomorphism of the image domain. We introduce SeqVP, built around this certificate. A proximal Jacobian-determinant-curl formulation yields inner problems whose penalties on changes in the map, Jacobian determinant, and curl vanish to first order at the current iterate. The method decreases a fixed outer energy up to a floating-point resolution guard and certifies every committed iterate. After certified initialization, stopping at any time returns a folding-free map. A Poisson-preconditioned gradient method with Jacobian-aware step prediction and affine-scaling damping, initialized by certified handoff from an uncertified binary32 pre-optimizer, handles a 1 mm brain MRI with about 2e7 unknowns in 90 s on one GPU. On 49 held-out OASIS pairs, SeqVP achieves mean Dice 0.821 over 35 structures and 0.889 over 4 tissue classes, with certified zero folding. Gains are 0.022 and 0.025 over ANTs SyN, and 0.022 and 0.023 over uniGradICON with instance optimization at equal time (paired, Holm-adjusted p < 0.001), but maps are rougher. SeqVP is level with ANTs on LPBA40, behind it on IBSR18, and ahead of all label-free baselines on 20 large-deformation pairs. On fields from nine methods, the certificate finds non-positive-Jacobian cells in every ANTs SyN map (median 10, missed by voxelwise checks) and in maps from NiftyReg, ConvexAdam, uniGradICON, and TransMorph.

cs.GR · math.AP

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05130

Solving the Nonlinear Westervelt Equation in Ultrasound Propagation Using Physics-Informed Neural Networks

Luke Oluwaseye Joel

原文摘要

Modeling the nonlinear acoustic wave propagation in biological tissues needs accuracy, especially in ultrasound imaging and therapeutic applications. The 1D Westervelt equation, \( p_{xx} - p_{tt} + γ\, p_{ttt} + η\,(p^{2})_{tt}=0, \) could be used to describe a finite-amplitude ultrasound propagation in these tissues. This equation accounts for nonlinearity, \(η\), and attenuation, \(γ\). To solve this equation, we observe that traditional solvers may struggle to do so correctly. This is due to the difficulties arising from the nonlinear nature of the equation. We solve the equation by defining three synthetic cases - linear (\(η=0.001\), \(γ=10^{-6}\)), shock-dominated (\(η=0.02\), \(γ=10^{-6}\)), and dissipation-dominated (\(η=0.01\), \(γ=5\times10^{-4}\)) - and using Physics-Informed Neural Networks (PINNs) to showcase the solution. A sinusoidal activation function is used to capture the oscillatory behavior of the problem. The results of our study, with PDE residual converging to at least \(10^{-6}\), boundary condition loss to \(10^{-5}\), and the PINN loss converging to \(10^{-5}\) for each case at \(30,000\) epochs, show that \(γ\) scales the rate of amplitude decay, while \(η\) governs frequency-dependent phase speeds.

math.AP · cs.LG · cs.NE

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04955

Deep Neural Operator Learning for stochastic heat equations and applications

Hengrong Du, Qi Feng, Weinan Wang

原文摘要

In this paper, we establish a universal approximation theorem for solution operators of semilinear stochastic heat equations in spatial dimension \(d\ge2\), extending deep neural operator approximation results to SPDEs with distribution-valued Gaussian forcing. A smooth Fourier cutoff reduces the forcing to finitely many measurements, while Besov and Schauder estimates control the resulting solution error. The finite-dimensional solution map is then approximated by ReLU branch networks together with grid-based trunk functions, yielding explicit bounds on the numbers of forcing measurements and trunk functions and approximation in every finite stochastic moment. For linear equations, the reduction coincides with truncation of the first Wiener chaos. Lastly, as an application, we prove a high-probability local-in-time approximation theorem for stochastic Navier--Stokes equations, with finite measurements of both the initial condition and forcing and an explicit sensing rate.

cs.LG · math.AP · math.PR

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2609.14892

An explicit solution of the five-expert prediction PDE and the exact optimality set of COMB

Jeff Calder, Nadejda Drenska

原文摘要

In this paper, we derive an explicit solution of the stationary prediction with expert advice PDE for five experts. The formula is given in three regions. In the first two regions, it is the four-expert solution plus a single integral with an elementary positive density. In the third region, it is a finite sum of hyperbolic products whose coefficients are determined by one scalar quadrature. Our formula establishes that the rank ordered adversary $(1,0,1,0,0)$ achieves the Hamiltonian maximum in the PDE everywhere in the ordered sector, while the COMB strategy $(1,0,1,0,1)$ is optimal only on a lower dimensional subset of the sector (where $x_1=x_2$ and $x_3=x_4$). The verification of the Hamiltonian inequalities is a tedious task, part of which is completed with a computer assisted proof. The verification reduces to 21 scalar inequalities, which we prove using 147 exact rational Bernstein polynomial certificates. The exact certificates and their independent arithmetic checks are included in a supplement to this paper, and a Lean 4 formalization machine-checks the verification and both main theorems, apart from the viscosity characterization.

math.AP · cs.GT · cs.LG · math.OC

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2510.20094

On the Structure of Stationary Solutions to McKean-Vlasov Equations with Applications to Noisy Transformers

Krishnakumar Balasubramanian, Sayan Banerjee, Philippe Rigollet

原文摘要

We study stationary solutions of McKean-Vlasov equations on the circle. We establish an exact equivalence between solutions of the stationary McKean-Vlasov equation and an infinite-dimensional quadratic system of equations over Fourier coefficients, allowing explicit characterization of stationary states in a sequence space rather than a function space. This framework provides a transparent description of local bifurcations, their periodicity and resonance structures, while accommodating singular potentials. We derive analytic expressions for the emergence, form and shape (supercritical, critical, subcritical or transcritical) of bifurcations involving possibly multiple Fourier modes and connect them with discontinuous phase transitions. We also characterize, under suitable assumptions, the detailed structure of the stationary bifurcating solutions through leading small-amplitude coefficients for arbitrarily many fixed Fourier modes. At the global level, we establish regularity and concavity properties of the free energy landscape, proving existence, compactness, and coexistence of globally minimizing stationary measures, and identifying discontinuous phase transitions with points of non-differentiability of the minimum free energy map. We apply the theory to the Noisy Mean-Field Transformer model, where we show how changing the inverse temperature $β$ affects the geometry of the infinitely many bifurcations from the uniform measure. We also discuss how increasing $β$ motivates the study of approximate multi-mode stationary profiles as candidates for a separate metastability analysis; persistence times require additional dynamical estimates. Moreover, we determine the exact threshold $β_0\simeq2.447$ separating continuous and discontinuous phase transitions in this model. The same threshold marks the change of the first bifurcation from supercritical to subcritical.

math.PR · cs.AI · cs.LG · math.AP · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.10277

Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs

Koichi Taniguchi, Sho Sonoda

原文摘要

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

cs.LG · math.AP · stat.ML

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2604.08095

The estimates for Boolean surface area of polynomial threshold functions

Fan Chang, Joseph Slote, Chun-Kai Tseng, Alexander Volberg, Haonan Zhang

原文摘要

Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture recently solved by OpenAI, \cite{GLsol}. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the second bound \[ \mathbf{BSA}[f]\le \exp(C(d)\sqrt{\log n}). \] This bound was obtained in \cite{FSVZ} without clarifying the magnitude of $C(d)$, which we do below. For lower moments $\bE |\nabla f|^η \le C(d, η)$ there is no dependence on $n$ for $0<η<1$.

cs.CC · math.AP · math.CA · math.PR

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arXiv ↗PDF ↗