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Takashi Furuya

Publications and source records attributed to Takashi Furuya.

At least 19 recordsLinked to original sources
arXiv2026

Mixtures of Neural Operators Reduce Active Complexity in Operator Learning

Operator-learning systems are not governed solely by total parameter count; for one query, the relevant bottleneck can be the model that must be loaded and evaluated. We study this distinction for classical neural operators on compact Sobolev subsets through a constructive comparison between routed mixtures of neural operators (MoNOs) and a fixed single-neural-operator construction. The comparison concerns expert-active complexity relative to that baseline, with total stored size and routing search accounted separately. A MoNO routes each input function through a tree to one expert. Our main theorem shows that every scalar uniformly continuous nonlinear operator with bounded output Sobolev radius on the approximation set admits a MoNO approximation whose active expert has smaller depth, width, and rank scaling than the analyzed single-neural-operator construction; for Lipschitz targets these expert quantities are bounded by $\mathcal{O}(\varepsilon^{-1})$. The theorem turns localization into an operator-level accounting of active expert size, routing depth, and number of experts. We also prove a quantitative universal approximation theorem for the underlying neural-operator architecture, with explicit dependence on compact-set diameter and modulus of continuity.

cs.LG↗
arXiv2026

Function graph transformers universally approximate operators between function spaces

We study the approximation of nonlinear operators between function spaces by transformers. Our approach is to lift functions to measures supported on their graphs and leverage a recently introduced measure-theoretic view of transformers. A function $h$ is represented by its graph measure $γ_h$, with finite tokens $\{(x_j,h(x_j))\}_{j=1}^N$ being its empirical approximations. We show that this framework elegantly models discretization refinement via convergence of measures and provides a natural setting for operator learning. Within this framework, we introduce function graph transformers, a graph-preserving subclass of measure-theoretic transformers that maps graph measures to graph measures, which is to say that outputs remain single-valued functions. Crucially, this additional structure does not reduce generality: we prove that the resulting graph-preserving maps can be approximated by finite compositions of standard softmax self-attention layers and pointwise MLPs, yielding universal approximation results for broad classes of nonlinear operators. Unlike existing theoretical approaches to operator learning with transformers, the measure-theoretic framework also accommodates regularized negative-order Sobolev inputs for which discretization invariance is particularly challenging, as well as query points on different output domains. Overall, function graph transformers provide a continuum viewpoint and mathematical toolkit for transformer-based operator learning, clarifying the roles of positional encodings, graph structure, regularization, and ensuring consistency across discretizations.

cs.LG↗
arXiv2026

Training Infinitely Deep and Wide Transformers

Transformers have become the dominant architecture in modern machine learning, yet the theoretical understanding of their training dynamics remains limited. This paper develops a rigorous mathematical framework for analyzing gradient-based training of transformers in the mean-field regime, where both the depth (number of layers) and width (number of attention heads) tend to infinity. While ResNet training can be understood as controlling a neural ODE, transformer training corresponds to controlling a neural PDE, due to the coupling of multiple token distributions through the attention mechanism. Our mean-field model features two types of measure representations: token distributions evolving through layers and attention parameters at each layer. We establish well-posedness of the forward pass through infinitely deep transformers, characterizing token evolution via flow maps that satisfy ODEs in function spaces. Using adjoint sensitivity analysis, we derive an explicit formula for the conditional Wasserstein gradient of the training risk, involving adjoint variables governed by backward ODEs. We prove the existence and uniqueness of gradient flow curves in the conditional Wasserstein metric space, establishing a rigorous foundation for gradient-based transformer training. A key technical contribution is providing necessary and sufficient conditions for injectivity of the Neural Tangent Kernel (NTK) for attention mechanisms: we show that NTK injectivity is equivalent to linear independence of log-sum-exp functions modulo affine functions, a condition satisfied by diverse token distributions, including discrete distributions, uniform distributions, and Gaussian mixtures. Under this NTK injectivity assumption, we prove that gradient flow converges to global minima when the initial loss is sufficiently small, eliminating spurious local minima from the optimization landscape.

math.OC↗
arXiv2026

Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.

cs.LG↗
arXiv2026

Approximation of Maximally Monotone Operators : A Graph Convergence Perspective

Operator learning has been highly successful for continuous mappings between infinite-dimensional spaces, such as PDE solution operators. However, many operators of interest-including differential operators-are discontinuous or set-valued, and lie outside classical approximation frameworks. We propose a paradigm shift by formulating approximation via graph convergence (Painlevé-Kuratowski convergence), which is well-suited for closed operators. We show that uniform and $L^p$ approximation are fundamentally inadequate in this setting. Focusing on maximally monotone operators, we prove that any such operator can be approximated in the sense of local graph convergence by continuous encoder-decoder architectures, and further construct structure-preserving approximations that retain maximal monotonicity via resolvent-based parameterizations.

cs.LG↗
arXiv2026

Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs

Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they do not address \emph{complexity}: for broad operator classes described only through regularity (e.g.\ uniform continuity or $C^r$-regularity), information-theoretic lower bounds imply that minimax-optimal NO approximation rates scale \emph{exponentially} in the reciprocal accuracy $1/\varepsilon$. This has shifted the focus of NO theory toward identifying additional problem-specific structure, beyond regularity, under which suitably tailored NO architectures can leverage to unlock polynomial scaling in $1/\varepsilon$. We exhibit the first polynomial-scaling regime for NO approximations of solution operators in stochastic analysis; by identifying structured families of \emph{non-Markovian} BSDEs with randomized terminal condition parameterized by the Sobolev-regular terminal condition and by Sobolev-regular additive nonlinear perturbations of the generator. We prove that their solution operator can be approximated (uniformly over the family) by a tailored NO whose number of trainable parameters grows \emph{polynomially} in $1/\varepsilon$. We unlock this polynomial scaling regime by \emph{informing the NO's inductive bias} by factoring out the singular part of the associated semilinear elliptic PDE Green's function and by incorporating the Doléans--Dade exponential of the BSDE's common non-Markovian factor into the NO's decoding layers. As a byproduct, we extend polynomial-scaling guarantees from families of linear elliptic PDEs on regular domains to the semilinear setting.

math.OC↗
arXiv2026

Approximation Theory for Lipschitz Continuous Transformers

Stability and robustness are critical for deploying Transformers in safety-sensitive settings. A principled way to enforce such behavior is to constrain the model's Lipschitz constant. However, approximation-theoretic guarantees for architectures that explicitly preserve Lipschitz continuity have yet to be established. In this work, we bridge this gap by introducing a class of gradient-descent-type in-context Transformers that are Lipschitz-continuous by construction. We realize both MLP and attention blocks as explicit Euler steps of negative gradient flows, ensuring inherent stability without sacrificing expressivity. We prove a universal approximation theorem for this class within a Lipschitz-constrained function space. Crucially, our analysis adopts a measure-theoretic formalism, interpreting Transformers as operators on probability measures, to yield approximation guarantees independent of token count. These results provide a rigorous theoretical foundation for the design of robust, Lipschitz continuous Transformer architectures.

cs.LG↗
arXiv2025

One model to solve them all: 2BSDE families via neural operators

We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees.

cs.LG↗
arXiv2025

Approximation theory for 1-Lipschitz ResNets

1-Lipschitz neural networks are fundamental for generative modelling, inverse problems, and robust classifiers. In this paper, we focus on 1-Lipschitz residual networks (ResNets) based on explicit Euler steps of negative gradient flows and study their approximation capabilities. Leveraging the Restricted Stone-Weierstrass Theorem, we first show that these 1-Lipschitz ResNets are dense in the set of scalar 1-Lipschitz functions on any compact domain when width and depth are allowed to grow. We also show that these networks can exactly represent scalar piecewise affine 1-Lipschitz functions. We then prove a stronger statement: by inserting norm-constrained linear maps between the residual blocks, the same density holds when the hidden width is fixed. Because every layer obeys simple norm constraints, the resulting models can be trained with off-the-shelf optimisers. This paper provides the first universal approximation guarantees for 1-Lipschitz ResNets, laying a rigorous foundation for their practical use.

cs.LG↗
arXiv2025

Transformers through the lens of support-preserving maps between measures

Transformers are deep architectures that define ``in-context maps'' which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for a vision transformer). In previous work, we studied the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically, uniformly analyze their expressivity, we considered the case that the mappings are conditioned on a context represented by a probability distribution which becomes discrete for a finite number of tokens. Modeling neural networks as maps on probability measures has multiple applications, such as studying Wasserstein regularity, proving generalization bounds and doing a mean-field limit analysis of the dynamics of interacting particles as they go through the network. In this work, we study the question what kind of maps between measures are transformers. We fully characterize the properties of maps between measures that enable these to be represented in terms of in-context maps via a push forward. On the one hand, these include transformers; on the other hand, transformers universally approximate representations with any continuous in-context map. These properties are preserving the cardinality of support and that the regular part of their Fréchet derivative is uniformly continuous. Moreover, we show that the solution map of the Vlasov equation, which is of nonlocal transport type, for interacting particle systems in the mean-field regime for the Cauchy problem satisfies the conditions on the one hand and, hence, can be approximated by a transformer; on the other hand, we prove that the measure-theoretic self-attention has the properties that ensure that the infinite depth, mean-field measure-theoretic transformer can be identified with a Vlasov flow.

cs.CL↗
arXiv2025

Approximation Rates in Besov Norms and Sample-Complexity of Kolmogorov-Arnold Networks with Residual Connections

Inspired by the Kolmogorov-Arnold superposition theorem, Kolmogorov-Arnold Networks (KANs) have recently emerged as an improved backbone for most deep learning frameworks, promising more adaptivity than their multilayer perceptron (MLP) predecessor by allowing for trainable spline-based activation functions. In this paper, we probe the theoretical foundations of the KAN architecture by showing that it can optimally approximate any Besov function in $B^{s}_{p,q}(\mathcal{X})$ on a bounded open, or even fractal, domain $\mathcal{X}$ in $\mathbb{R}^d$ at the optimal approximation rate with respect to any weaker Besov norm $B^α_{p,q}(\mathcal{X})$; where $α< s$. We complement our approximation result with a statistical guarantee by bounding the pseudodimension of the relevant class of Res-KANs. As an application of the latter, we directly deduce a dimension-free estimate on the sample complexity of a residual KAN model when learning a function of Besov regularity from $N$ i.i.d. noiseless samples, showing that KANs can learn the smooth maps which they can approximate.

cs.LG↗
arXiv2025

Is In-Context Universality Enough? MLPs are Also Universal In-Context

The success of transformers is often linked to their ability to perform in-context learning. Recent work shows that transformers are universal in context, capable of approximating any real-valued continuous function of a context (a probability measure over $\mathcal{X}\subseteq \mathbb{R}^d$) and a query $x\in \mathcal{X}$. This raises the question: Does in-context universality explain their advantage over classical models? We answer this in the negative by proving that MLPs with trainable activation functions are also universal in-context. This suggests the transformer's success is likely due to other factors like inductive bias or training stability.

stat.ML↗
arXiv2025

Theoretical Error Analysis of Entropy Approximation for Gaussian Mixtures

Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated analytically. In this paper, we study the approximate entropy represented as the sum of the entropies of unimodal Gaussian distributions with mixing coefficients. This approximation is easy to calculate analytically regardless of dimension, but there is a lack of theoretical guarantees. We theoretically analyze the approximation error between the true and the approximate entropy to reveal when this approximation works effectively. This error is essentially controlled by how far apart each Gaussian component of the Gaussian mixture is. To measure such separation, we introduce the ratios of the distances between the means to the sum of the variances of each Gaussian component of the Gaussian mixture, and we reveal that the error converges to zero as the ratios tend to infinity. In addition, the probabilistic estimate indicates that this convergence situation is more likely to occur in higher-dimensional spaces. Therefore, our results provide a guarantee that this approximation works well for high-dimensional problems, such as neural networks that involve a large number of parameters.

stat.ML↗
arXiv2024

Can neural operators always be continuously discretized?

We consider the problem of discretization of neural operators between Hilbert spaces in a general framework including skip connections. We focus on bijective neural operators through the lens of diffeomorphisms in infinite dimensions. Framed using category theory, we give a no-go theorem that shows that diffeomorphisms between Hilbert spaces or Hilbert manifolds may not admit any continuous approximations by diffeomorphisms on finite-dimensional spaces, even if the approximations are nonlinear. The natural way out is the introduction of strongly monotone diffeomorphisms and layerwise strongly monotone neural operators which have continuous approximations by strongly monotone diffeomorphisms on finite-dimensional spaces. For these, one can guarantee discretization invariance, while ensuring that finite-dimensional approximations converge not only as sequences of functions, but that their representations converge in a suitable sense as well. Finally, we show that bilipschitz neural operators may always be written in the form of an alternating composition of strongly monotone neural operators, plus a simple isometry. Thus we realize a rigorous platform for discretization of a generalization of a neural operator. We also show that neural operators of this type may be approximated through the composition of finite-rank residual neural operators, where each block is strongly monotone, and may be inverted locally via iteration. We conclude by providing a quantitative approximation result for the discretization of general bilipschitz neural operators.

cs.LG↗
arXiv2024

Transformers are Universal In-context Learners

Transformers are deep architectures that define "in-context mappings" which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for a vision transformer). In this work, we study in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically, uniformly address their expressivity, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens which becomes discrete for a finite number of these. The relevant notion of smoothness then corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLPs between multi-head attention layers is also explicitly controlled. We consider both unmasked attentions (as used for the vision transformer) and masked causal attentions (as used for NLP and time series applications). We tackle the causal setting leveraging a space-time lifting to analyze causal attention as a mapping over probability distributions of tokens.

cs.CL↗
arXiv2024

Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations

Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding integral equations via Duahamel's principle, and to leverage the similarity between neural operators and Picard's iteration, a classical algorithm for solving PDEs. This approach is potentially generalizable beyond parabolic PDEs to a range of other equations, including the Navier-Stokes equation, nonlinear Schrödinger equations and nonlinear wave equations, which can be solved by Picard's iteration.

cs.LG↗
arXiv2024

Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts

Mixture-of-Experts (MoEs) can scale up beyond traditional deep learning models by employing a routing strategy in which each input is processed by a single "expert" deep learning model. This strategy allows us to scale up the number of parameters defining the MoE while maintaining sparse activation, i.e., MoEs only load a small number of their total parameters into GPU VRAM for the forward pass depending on the input. In this paper, we provide an approximation and learning-theoretic analysis of mixtures of expert MLPs with (P)ReLU activation functions. We first prove that for every error level $\varepsilon>0$ and every Lipschitz function $f:[0,1]^n\to \mathbb{R}$, one can construct a MoMLP model (a Mixture-of-Experts comprising of (P)ReLU MLPs) which uniformly approximates $f$ to $\varepsilon$ accuracy over $[0,1]^n$, while only requiring networks of $\mathcal{O}(\varepsilon^{-1})$ parameters to be loaded in memory. Additionally, we show that MoMLPs can generalize since the entire MoMLP model has a (finite) VC dimension of $\tilde{O}(L\max\{nL,JW\})$, if there are $L$ experts and each expert has a depth and width of $J$ and $W$, respectively.

stat.ML↗
arXiv2023

Convergences for Minimax Optimization Problems over Infinite-Dimensional Spaces Towards Stability in Adversarial Training

Training neural networks that require adversarial optimization, such as generative adversarial networks (GANs) and unsupervised domain adaptations (UDAs), suffers from instability. This instability problem comes from the difficulty of the minimax optimization, and there have been various approaches in GANs and UDAs to overcome this problem. In this study, we tackle this problem theoretically through a functional analysis. Specifically, we show the convergence property of the minimax problem by the gradient descent over the infinite-dimensional spaces of continuous functions and probability measures under certain conditions. Using this setting, we can discuss GANs and UDAs comprehensively, which have been studied independently. In addition, we show that the conditions necessary for the convergence property are interpreted as stabilization techniques of adversarial training such as the spectral normalization and the gradient penalty.

stat.ML↗