Selected work

Representative Papers

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×.

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RRaPINNs: Residual Risk-Aware Physics Informed Neural Networks

Nov 23, 2025

Physics-informed neural networks (PINNs) minimize the mean residual, which can obscure localized large errors and compromise solution reliability. To address this, we propose a residual risk-aware training framework that explicitly targets tail residual control. Our method introduces conditional value-at-risk (CVaR) and mean excess (ME) surrogate penalties, establishing a theoretical connection between risk-sensitive optimization and chance-constrained learning. We further design three mechanisms: persistent hard-point replay, local risk budgeting, and multi-objective risk modeling—enabling explicit trade-offs between accuracy and tail robustness. Experiments on Burgers, heat, Korteweg–de Vries (KdV), and Poisson equations demonstrate that our approach reduces tail errors by up to 57%, maintains or improves mean error, enhances training stability, and effectively handles discontinuous solutions.

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Recent publications

Latest Papers

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

RRaPINNs: Residual Risk-Aware Physics Informed Neural Networks

Nov 23, 2025

Physics-informed neural networks (PINNs) minimize the mean residual, which can obscure localized large errors and compromise solution reliability. To address this, we propose a residual risk-aware training framework that explicitly targets tail residual control. Our method introduces conditional value-at-risk (CVaR) and mean excess (ME) surrogate penalties, establishing a theoretical connection between risk-sensitive optimization and chance-constrained learning. We further design three mechanisms: persistent hard-point replay, local risk budgeting, and multi-objective risk modeling—enabling explicit trade-offs between accuracy and tail robustness. Experiments on Burgers, heat, Korteweg–de Vries (KdV), and Poisson equations demonstrate that our approach reduces tail errors by up to 57%, maintains or improves mean error, enhances training stability, and effectively handles discontinuous solutions.

0 citationsRead paper