# Taylor-Structured Online Adapter

- ID: 3099
- Canonical URL: https://synthcore.org/idea/3099/taylor-structured-online-adapter
- API JSON: https://synthcore.org/api/idea/3099.json
- API Markdown: https://synthcore.org/api/idea/3099.md
- Verification status: unverified
- Source: [arXiv:2609.03348](https://arxiv.org/abs/2609.03348)
- Category: training
- Solves: sample-efficiency, accuracy, stability
- ML areas: world-model, ssm, rnn, fine-tuning, training-dynamics
- Math tags: dynamical-systems, control-theory, statistics, approximation-theory, linear-algebra
- Ratings: usefulness 6/10; difficulty 4/10; novelty 6/10

## Idea description

Attach a small, online-adapted Taylor-polynomial residual model to a neural dynamics model instead of adapting all network weights. Construct the dictionary from the local Taylor structure, remove monomials forbidden by known joint-odd symmetry, and update its coefficients with recursive least squares. This yields fast adaptation with an interpretable parameter count and a direct test of whether the assumed symmetry is valid.

## Mathematical statement

Let $z=(x,u)$ be the concatenated state-input vector and let $\varphi_D^{\mathrm{FE}}(z)$ contain retained monomials through total degree $D$ selected from the forward-Euler/Taylor structure. The identified map is $\widehat{F}_{D,k}(z)=\widehat{\Theta}_{D,k}^{\top}\varphi_D^{\mathrm{FE}}(z)$, where $\widehat{\Theta}_{D,k}$ contains one coefficient vector per output dimension. For a measured transition $(z_k,x_{k+1})$, matrix RLS uses gain $K_k$, covariance $P_k$, forgetting factor $0<\lambda\le1$, and coefficient residual $e_k$. Joint-odd symmetry retains monomials with odd total degree and removes even-total-degree monomials; additional forward-Euler pruning removes interactions absent from the known local vector field.

## Key formulas

- $$\widehat{F}_{D,k}(x,u)=\widehat{\Theta}_{D,k}^{\top}\varphi_D^{\mathrm{FE}}(x,u),$$
- $$K_k=\frac{P_k\varphi_k}{\lambda+\varphi_k^{\top}P_k\varphi_k},\qquad \widehat{\Theta}_{k+1}=\widehat{\Theta}_k+(x_{k+1}-\widehat{\Theta}_k^{\top}\varphi_k)K_k^{\top},$$
- $$P_{k+1}=\lambda^{-1}\left(P_k-K_k\varphi_k^{\top}P_k\right),\qquad 0<\lambda\le1.$$
- $$\varphi_D^{\mathrm{odd}}(z)=\{z^\alpha:\ |\alpha|\le D,\ |\alpha|\ \mathrm{odd}\},$$

## Implementation notes

Use this as a lightweight adaptation head on a pretrained neural state-space or world model. Freeze the base network $h_\theta(z)$ and predict $x_{k+1}=h_\theta(z_k)+R_k^T\varphi_D^{\mathrm{FE}}(z_k)$, where $R_k$ is updated online by RLS; alternatively let the polynomial head replace the final transition layer. Build the monomial list once, remove even-total-degree monomials only when the environment has verified joint-odd symmetry, and apply forward-Euler pruning only when the corresponding interaction is physically absent. Pseudocode: initialize $R=0$, $P=\alpha I$; at each transition compute $\varphi_k$, residual $e_k=x_{k+1}-h_\theta(z_k)-R^T\varphi_k$, gain $K_k=P\varphi_k/(\lambda+\varphi_k^TP\varphi_k)$, update $R\leftarrow R+e_kK_k^T$, and update $P\leftarrow\lambda^{-1}(P-K_k\varphi_k^TP)$. The paper provides the Taylor-informed reduced dictionary and online RLS; the neural residual-head construction is the transfer. Compute covariance from $P$ and damp or reject updates when $\varphi_k^TP\varphi_k$ is excessively large. The first cheap experiment is a regime-shifted Lorenz-63 or nonlinear-pendulum dataset: pretrain on one parameter range, shift the parameter online, and compare full fine-tuning, unstructured polynomial RLS, and the symmetry/pruned adapter. The prediction is that, when symmetry is correct, even-degree coefficient estimates remain statistically indistinguishable from zero and the reduced adapter reaches a target one-step error with fewer samples. After a regime shift, adaptation error should decrease approximately geometrically at a rate controlled by the forgetting factor $\lambda$ until reaching a persistent-excitation noise floor.

## Disclaimer

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