# AAA-Rational Coordinate Trunk

- ID: 2580
- Canonical URL: https://synthcore.org/idea/2580/aaa-rational-coordinate-trunk
- API JSON: https://synthcore.org/api/idea/2580.json
- API Markdown: https://synthcore.org/api/idea/2580.md
- Verification status: unverified
- Source: [arXiv:2608.19658](https://arxiv.org/abs/2608.19658)
- Category: architecture
- Solves: accuracy, stability, scalability
- ML areas: mlp, diffusion, graph-nn
- Math tags: approximation-theory, numerical-analysis, linear-algebra, pde
- Ratings: usefulness 6/10; difficulty 5/10; novelty 6/10

## Idea description

Replace or augment the coordinate embedding of a neural operator, PINN, or coordinate MLP with Chebyshev features plus rational features whose poles are selected by the AAA rational approximation algorithm. The rational features should represent boundary layers and other localized singular structures with fewer channels than a high-degree polynomial basis, reducing Gibbs-like oscillations and improving accuracy at small diffusion-to-advection ratios.

## Mathematical statement

The benchmark equation is the singularly perturbed boundary-value problem $-\varepsilon u''(x)+u'(x)+u(x)=q(x)$ for $x\in(0,1)$, with $u(0)=u(1)=0$, where $\varepsilon>0$ is the diffusion-to-advection ratio, $q(x)$ is a source function, and $u(x)$ is the solution. The transferred representation is a trunk basis consisting of Chebyshev polynomials $T_k(\xi)=\cos(k\arccos \xi)$ on the rescaled coordinate $\xi=2x-1$, together with rational functions in barycentric form $r_j(x)=\frac{\sum_{i=1}^{m_j}w_{j,i}f_i/(x-z_{j,i})}{\sum_{i=1}^{m_j}w_{j,i}/(x-z_{j,i})}$. Here $z_{j,i}$ are sampled support points, $f_i$ are sampled values of a target profile or residual, and $w_{j,i}$ are AAA barycentric weights. AAA chooses the weights as a nonzero right singular vector associated with the smallest singular value of the Loewner matrix $L_{\ell i}=(f_\ell-f_i)/(z_\ell-z_i)$ over nonsupport points $z_\ell$ and support points $z_i$, then adds the point with largest current approximation error. For a neural trunk, use the resulting poles $p_j$ and residues or barycentric coefficients as fixed features, for example $\rho_j(x)=1/(x-p_j)$ or the stabilized barycentric $r_j(x)$. Poles should be kept off the real domain, typically $p_j=a_j+ib_j$ with $b_j\neq0$, so the features remain bounded on $[0,1]$. The model output is $\hat u(x;q)=\sum_{k=0}^{K}b_k(q)T_k(2x-1)+\sum_{j=1}^{J}c_j(q)\rho_j(x)$, where the branch network produces coefficients $b_k(q),c_j(q)$ from the input function or parameterized forcing.

## Key formulas

- $$-\varepsilon\frac{d^{2}u}{dx^{2}}(x)+\frac{du}{dx}(x)+u(x)=q(x),\qquad x\in(0,1),$$
- $$u(0)=u(1)=0.$$
- $$r(x)=\frac{\displaystyle\sum_{i=1}^{m}\frac{w_i f_i}{x-z_i}}{\displaystyle\sum_{i=1}^{m}\frac{w_i}{x-z_i}},\qquad L_{\ell i}=\frac{f_\ell-f_i}{z_\ell-z_i},\qquad w=\operatorname{sv}_{\min}(L),$$
- $$\hat u(x;q)=\sum_{k=0}^{K}b_k(q)T_k(2x-1)+\sum_{j=1}^{J}c_j(q)\frac{1}{x-p_j},\qquad T_k(\xi)=\cos(k\arccos\xi).$$

## Implementation notes

1. Integration point: use this as the trunk of a DeepONet or as the coordinate encoder of a coordinate MLP. The branch input is a discretized forcing function, boundary condition, or conditioning vector; the trunk input is a scalar coordinate $x$ or one coordinate per spatial query. Concatenate normalized Chebyshev features $[T_0(2x-1),\ldots,T_K(2x-1)]$ with rational features generated from poles near expected layers. 2. MVP pseudocode: choose a small calibration set of training solutions or residual profiles; sample $x_n\in[0,1]$; initialize support points; repeatedly form $L_{\ell i}=(f_\ell-f_i)/(x_\ell-x_i)$, compute its smallest right singular vector $w$, evaluate the barycentric rational approximation, add the point with largest absolute error, and stop at $J$ terms or a tolerance. Convert each selected complex pole to real features $\operatorname{Re}(1/(x-p_j))$ and $\operatorname{Im}(1/(x-p_j))$, or directly use stabilized barycentric rational functions. During training compute $b(q)$ and $c(q)$ with the branch network and contract them with the trunk features. 3. Mathematics versus estimation: the Chebyshev recurrence and barycentric formula are exact; support points, poles, and weights are estimated once from calibration profiles. If no full solutions are available, run AAA on PDE residual snapshots, high-gradient training targets, or an analytic proxy boundary layer $f(x)=\exp(-(1-x)/\varepsilon)$. Clip $|x-p_j|$ below a small threshold and regularize coefficient norms. 4. First experiment: solve the stated BVP for random smooth $q$ at $\varepsilon\in\{10^{-2},10^{-3},10^{-4}\}$ using a small DeepONet with fixed trunk width 32. Compare a vanilla MLP, a degree-31 Chebyshev trunk, and a 12-polynomial plus 10-complex-pole rational trunk at equal parameter count. Measure relative $L^2$ error, maximum error in the last $5\%$ of the domain, and oscillation count near the layer. Success is lower boundary-layer error and smoother predictions at equal FLOPs, especially as $\varepsilon$ decreases; a useful target is at least 20% lower maximum near-wall error without increasing trunk width.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.