Selected work

Representative Papers

MiNO: Cotangent-bundle propagator learning for PDEs

Aug 15, 2026

This study addresses the regularity loss caused by discontinuities in PDE solving by proposing to learn phase-space propagators rather than directly fitting solution fields. By modeling phase and amplitude via eikonal and transport equations and reconstructing solutions through oscillatory integrals, the method transforms singularity propagation into a geometric problem, effectively decoupling it from field approximation. In discontinuous advection benchmarks, this approach rapidly converges to theoretical accuracy limits. Furthermore, it outperforms Fourier Neural Operators in smooth regimes and demonstrates robust generalization across multiple initial conditions with a single model. Consequently, this framework successfully overcomes the accuracy bottlenecks of traditional neural operators when handling non-smooth solutions.

0 citationsRead paper

SEAM: Global consistency beyond local accuracy in scientific machine learning

Aug 06, 2026

In scientific machine learning, local explanations, while often accurate, frequently lack global consistency. To address this, this work proposes the SEAM framework—the first approach to enable computable global explanation consistency across regions, sensors, and model components. SEAM models explanations via an explanation-sheaf formalism that integrates system states, closures, observation channels, and contract metadata. By leveraging channel-wise parsing obstructions and residual-aware regularization, it structurally assembles local explanations into globally coherent scientific interpretations, explicitly distinguishing between inconsistency and non-identifiability. Evaluated across 19 experiments, SEAM successfully identifies scenarios where local predictions are accurate yet explanations conflict, precisely attributing discrepancies to specific channels and overlapping regions, thereby supporting generator monitoring under distributional shift.

0 citationsRead paper

AXLE: A Cloud Infrastructure for Lean 4 Theorem Proving Utilities

Jun 24, 2026

Existing infrastructure struggles to meet the demands of AI-driven mathematical research for Lean 4, particularly in high-throughput processing, scalable verification, multi-version support, and request-level isolation. This work proposes the first cloud-native Lean 4 service platform, which uniquely enables high concurrency, per-request isolation, and coexistence of multiple Lean 4 and Mathlib versions. The platform integrates 14 metaprogramming tools—including proof checking, semantic source code manipulation, deterministic repair, and lemma extraction—and provides seamless access via HTTP API, Python SDK, CLI, and a web UI, eliminating the need for local deployment. Already publicly deployed, it has processed over 500 million requests and powered Axiom Math’s perfect score in the 2025 Putnam Competition, thereby addressing a critical gap in scalable theorem-proving infrastructure.

0 citationsRead paper

FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

Mar 30, 2026

This work addresses the limitations of traditional Kolmogorov–Arnold Networks (KANs), which rely on fixed-grid B-spline bases and struggle to approximate nonsmooth or multiscale functions effectively. The authors introduce, for the first time, fractal geometry into neural function approximation by proposing Pure FI-KAN and Hybrid FI-KAN architectures. These models employ learnable fractal interpolation function (FIF) bases constructed via iterated function systems (IFS) and incorporate a differentiable fractal dimension as an adaptive parameter to dynamically align the Hölder regularity of the basis functions with that of the target function. Empirical results demonstrate substantial improvements over conventional KANs: on Hölder continuity benchmarks, performance gains range from 1.3× to 33×; mean squared error on fractal targets is reduced by 6.3×; and accuracy in approximating solutions to nonsmooth partial differential equations improves by up to 79×.

0 citationsRead paper

Müntz-Szász Networks: Neural Architectures with Learnable Power-Law Bases

Dec 22, 2025

Many physical models exhibit singularities—e.g., boundary layers or $|x|^alpha$-type behavior in fracture mechanics—where standard neural networks, constrained by fixed activation functions, suffer from poor approximation efficiency. To address this, we propose a learnable power-law basis network, the first architecture to explicitly embed the Müntz–Szász approximation theorem. Its activation function, $varphi(x)=sum a_k|x|^{mu_k}+sum b_koperatorname{sign}(x)|x|^{lambda_k}$, features jointly learnable exponents $mu_k,lambda_k$ and coefficients. We prove its universal approximation property and derive a squared-order error bound strictly superior to that of multilayer perceptrons (MLPs). Experiments demonstrate: (i) 5–8× lower error and 10× fewer parameters on singular regression tasks; (ii) 3–6× higher accuracy on physics-informed neural network (PINN) benchmarks; and (iii) learned exponents that quantitatively recover the true singularity structure of analytical solutions.

0 citationsRead paper

Recent publications

Latest Papers

MiNO: Cotangent-bundle propagator learning for PDEs

Aug 15, 2026

This study addresses the regularity loss caused by discontinuities in PDE solving by proposing to learn phase-space propagators rather than directly fitting solution fields. By modeling phase and amplitude via eikonal and transport equations and reconstructing solutions through oscillatory integrals, the method transforms singularity propagation into a geometric problem, effectively decoupling it from field approximation. In discontinuous advection benchmarks, this approach rapidly converges to theoretical accuracy limits. Furthermore, it outperforms Fourier Neural Operators in smooth regimes and demonstrates robust generalization across multiple initial conditions with a single model. Consequently, this framework successfully overcomes the accuracy bottlenecks of traditional neural operators when handling non-smooth solutions.

0 citationsRead paper

SEAM: Global consistency beyond local accuracy in scientific machine learning

Aug 06, 2026

In scientific machine learning, local explanations, while often accurate, frequently lack global consistency. To address this, this work proposes the SEAM framework—the first approach to enable computable global explanation consistency across regions, sensors, and model components. SEAM models explanations via an explanation-sheaf formalism that integrates system states, closures, observation channels, and contract metadata. By leveraging channel-wise parsing obstructions and residual-aware regularization, it structurally assembles local explanations into globally coherent scientific interpretations, explicitly distinguishing between inconsistency and non-identifiability. Evaluated across 19 experiments, SEAM successfully identifies scenarios where local predictions are accurate yet explanations conflict, precisely attributing discrepancies to specific channels and overlapping regions, thereby supporting generator monitoring under distributional shift.

0 citationsRead paper

AXLE: A Cloud Infrastructure for Lean 4 Theorem Proving Utilities

Jun 24, 2026

Existing infrastructure struggles to meet the demands of AI-driven mathematical research for Lean 4, particularly in high-throughput processing, scalable verification, multi-version support, and request-level isolation. This work proposes the first cloud-native Lean 4 service platform, which uniquely enables high concurrency, per-request isolation, and coexistence of multiple Lean 4 and Mathlib versions. The platform integrates 14 metaprogramming tools—including proof checking, semantic source code manipulation, deterministic repair, and lemma extraction—and provides seamless access via HTTP API, Python SDK, CLI, and a web UI, eliminating the need for local deployment. Already publicly deployed, it has processed over 500 million requests and powered Axiom Math’s perfect score in the 2025 Putnam Competition, thereby addressing a critical gap in scalable theorem-proving infrastructure.

0 citationsRead paper

FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

Mar 30, 2026

This work addresses the limitations of traditional Kolmogorov–Arnold Networks (KANs), which rely on fixed-grid B-spline bases and struggle to approximate nonsmooth or multiscale functions effectively. The authors introduce, for the first time, fractal geometry into neural function approximation by proposing Pure FI-KAN and Hybrid FI-KAN architectures. These models employ learnable fractal interpolation function (FIF) bases constructed via iterated function systems (IFS) and incorporate a differentiable fractal dimension as an adaptive parameter to dynamically align the Hölder regularity of the basis functions with that of the target function. Empirical results demonstrate substantial improvements over conventional KANs: on Hölder continuity benchmarks, performance gains range from 1.3× to 33×; mean squared error on fractal targets is reduced by 6.3×; and accuracy in approximating solutions to nonsmooth partial differential equations improves by up to 79×.

0 citationsRead paper

Müntz-Szász Networks: Neural Architectures with Learnable Power-Law Bases

Dec 22, 2025

Many physical models exhibit singularities—e.g., boundary layers or $|x|^alpha$-type behavior in fracture mechanics—where standard neural networks, constrained by fixed activation functions, suffer from poor approximation efficiency. To address this, we propose a learnable power-law basis network, the first architecture to explicitly embed the Müntz–Szász approximation theorem. Its activation function, $varphi(x)=sum a_k|x|^{mu_k}+sum b_koperatorname{sign}(x)|x|^{lambda_k}$, features jointly learnable exponents $mu_k,lambda_k$ and coefficients. We prove its universal approximation property and derive a squared-order error bound strictly superior to that of multilayer perceptrons (MLPs). Experiments demonstrate: (i) 5–8× lower error and 10× fewer parameters on singular regression tasks; (ii) 3–6× higher accuracy on physics-informed neural network (PINN) benchmarks; and (iii) learned exponents that quantitatively recover the true singularity structure of analytical solutions.

0 citationsRead paper