Tree-Based Polynomial Chaos Expansion

• Tree-PCE is a surrogate modeling technique that adaptively partitions the input space and applies local polynomial chaos expansions to capture discontinuities and nonlinearities.
• It improves on global PCE by fitting lower-degree surrogates in localized regions, reducing errors near sharp transitions and avoiding oscillations.
• The method facilitates global sensitivity analysis through analytical Sobol' indices and tree-induced variable importance measures derived from its adaptive structure.

Searching arXiv for the specified Tree-PCE paper to ground the article in the cited source. Tree-based Polynomial Chaos Expansion (Tree-PCE) is a piecewise surrogate modeling method that combines the analytical strengths of Polynomial Chaos Expansion (PCE) with an adaptive, tree-driven partition of the input space. It is intended for approximating complex model responses that are difficult for a single global polynomial to capture, especially when the output exhibits strong nonlinearities, threshold effects, discontinuities, or localized oscillations. The method recursively partitions the domain into hyperrectangular subdomains, fits a local PCE in each subdomain, and assembles the resulting local approximations into a piecewise polynomial surrogate. In the formulation introduced in "A tree-based Polynomial Chaos expansion for surrogate modeling and sensitivity analysis of complex numerical models" (Said et al., 16 Sep 2025), Tree-PCE is also designed to support global sensitivity analysis through both Sobol' indices derived from local expansion coefficients and a new class of indices induced by the tree structure itself.

1. Conceptual basis and relation to Polynomial Chaos Expansion

Standard PCE constructs a single global polynomial surrogate over the full input domain,

$$G(\mathbf{x}) \approx G^{\mathrm{PCE}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}} y_{\boldsymbol{\alpha}} \Psi_{\boldsymbol{\alpha}}(\mathbf{x}),$$

where $\Psi_{\boldsymbol{\alpha}}$ are orthonormal polynomial basis functions and the coefficients $y_{\boldsymbol{\alpha}}$ are estimated from data. Tree-PCE does not impose a single polynomial over the entire domain. Instead, it defines

$$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$

with

$$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$

This makes Tree-PCE a binary tree of local PCE surrogates. The central rationale is that local smoothness is easier to approximate than global irregularity. A global polynomial may struggle near discontinuities and sharp transitions because of the Gibbs phenomenon: increasing the polynomial degree may reduce global error but does not remove oscillations near jumps. Tree-PCE addresses this by isolating regions with different behavior; once a discontinuity is separated into different rectangles, each local PCE can operate on a smoother response and may remain at low degree (Said et al., 16 Sep 2025).

The paper contrasts this construction with standard or global PCE, sparse PCE, and Stochastic Spectral Embedding (SSE). Standard PCE uses one global polynomial, whereas Tree-PCE uses local polynomials. SSE also partitions the domain, but it expands residuals across multiple levels; Tree-PCE retains only terminal leaf models. The split-selection mechanism also differs: Tree-PCE scans candidate thresholds and minimizes global TSE, whereas SSE’s default median-based splitting can miss discontinuities more easily. This suggests that Tree-PCE is intended to align partitions more directly with nonsmooth features.

2. Mathematical formulation of local surrogates

Within each leaf rectangle $\mathcal{R}^r$, the local output approximation is expressed as

$$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}}\,\Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$

If the inputs are independent, the local multivariate basis is tensorized: $$\Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}) = \prod_{i=1}^d \phi^{(i),r}_{\alpha_i}(x_i).$$

The coefficients are estimated nonintrusively from samples in the corresponding region, typically by regression or sparse regression. The paper explicitly notes that the specific coefficient estimator is not essential to the Tree-PCE framework. This places the method at the level of a domain-decomposed surrogate architecture rather than a single prescribed estimation routine.

Because the surrogate is piecewise, the local basis, coefficient set, and even effective complexity may vary across regions. The representation therefore differs from a global polynomial expansion not only in support but also in locality of approximation. A plausible implication is that the method can allocate polynomial expressivity where needed while avoiding a globally high-order basis in regions where the target response is simpler. That implication is consistent with the paper’s emphasis on the compromise between the number of subdomains and the degree of local polynomials (Said et al., 16 Sep 2025).

3. Recursive partitioning and tree construction

The algorithm begins with the whole input rectangle $\mathcal{D}_{\mathbf{X}}$, on which a global PCE is first fitted. For each current rectangle $\mathcal{R}$, Tree-PCE evaluates candidate split thresholds along each input direction $\Psi_{\boldsymbol{\alpha}}$0. A threshold $\Psi_{\boldsymbol{\alpha}}$1 defines two subrectangles,

$\Psi_{\boldsymbol{\alpha}}$2

A local PCE is then fitted in each subdomain using the conditional distribution restricted to that rectangle.

Split selection is driven by the total squared error (TSE). For a rectangle $\Psi_{\boldsymbol{\alpha}}$3, the local error is

$\Psi_{\boldsymbol{\alpha}}$4

For each candidate split, the combined error of the two children is evaluated, and the algorithm chooses the split minimizing

$\Psi_{\boldsymbol{\alpha}}$5

Equivalently, it maximizes the gain

$\Psi_{\boldsymbol{\alpha}}$6

The rectangle with the largest positive gain is split next, yielding a binary tree. The process continues until no split improves the error or until a stopping criterion is met. A practical stopping rule introduced in the paper is

$\Psi_{\boldsymbol{\alpha}}$7

Under this rule, a split is accepted only if it improves the global error by a relative amount controlled by $\Psi_{\boldsymbol{\alpha}}$8. Small $\Psi_{\boldsymbol{\alpha}}$9 leads to finer partitions and higher accuracy, whereas larger $y_{\boldsymbol{\alpha}}$0 yields fewer classes and a simpler surrogate (Said et al., 16 Sep 2025).

The resulting tree is a binary decision structure. Each internal node stores the split direction $y_{\boldsymbol{\alpha}}$1 and threshold $y_{\boldsymbol{\alpha}}$2. Evaluation at a point $y_{\boldsymbol{\alpha}}$3 consists of traversing the tree from the root to a leaf via threshold comparisons, so evaluation cost is roughly proportional to the tree height rather than the number of rectangles. This operational property is one of the method’s stated computational advantages.

4. Sensitivity analysis: Sobol' indices and tree-induced indices

A major strength of PCE-type surrogates is that Sobol' indices can be derived analytically from polynomial coefficients. For independent inputs, the first-order and total Sobol' indices for standard PCE are given by

$y_{\boldsymbol{\alpha}}$4

and

$y_{\boldsymbol{\alpha}}$5

For Tree-PCE, analogous formulas are derived, but the piecewise structure introduces additional terms involving rectangle probabilities $y_{\boldsymbol{\alpha}}$6, partial-region probabilities $y_{\boldsymbol{\alpha}}$7, and overlap integrals

$y_{\boldsymbol{\alpha}}$8

The variance expression reported in the paper is

$y_{\boldsymbol{\alpha}}$9

The corresponding conditional variance $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$0 is also given analytically in Proposition 1 and involves sums over pairs of rectangles and basis functions.

The paper notes that these Tree-PCE Sobol' formulas are exact in principle but may become expensive when the number of classes $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$1 is large. In that regime, Tree-PCE can instead be used as a surrogate within a Monte Carlo pick-freeze procedure.

Beyond Sobol' analysis, the paper introduces a distinct tree-induced measure of variable importance: $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$2 where $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$3 is the initial global PCE error, $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$4 is the error reduction at split $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$5, $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$6 is the split direction at step $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$7, and $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$8 is the number of splits. Large values indicate that splitting along $$G^{\mathrm{Tree\text{-}PCE}}(\mathbf{x}) = \sum_{r=1}^{R} \mathbf{1}_{\mathbf{x}\in\mathcal{R}^r}\, G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}),$$9 was important for reducing surrogate error, while small values indicate that $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$0 is either less influential or already well captured by low-degree polynomials. These indices are described as free byproducts of Tree-PCE construction and are stated to work even for dependent inputs (Said et al., 16 Sep 2025).

5. Empirical behavior on synthetic and applied models

The paper reports numerical experiments on synthetic and real-world models, including a 2D morphodynamic case, to assess the balance between accuracy and complexity. In a simple piecewise-constant 2D example with a diagonal discontinuity, Tree-PCE first isolates the easy region and then refines near the diagonal discontinuity, yielding a piecewise approximation that matches the function structure much better than a global polynomial. This example is used to illustrate how adaptive rectangular splitting can identify sharp boundaries.

A higher-dimensional synthetic benchmark is also studied: $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$1 For the case $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$2, the response is smooth within each active region; Tree-PCE is reported to detect the discontinuities quickly, and after about 16 classes the discontinuity structure is resolved. In error-versus-complexity plots, Tree-PCE and sparse Tree-PCE outperform standard PCE, sparse PCE, and SSE. For $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$3, the response contains strong local oscillations; with low local degree, Tree-PCE has difficulty capturing the discontinuity, whereas increasing $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$4 improves performance and can reveal the split point. The stated trade-off is that better local polynomials require more samples, which can limit further partitioning. For $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$5, Tree-PCE still finds influential directions first; sparse Tree-PCE and sparse PCE perform best overall, with Tree-PCE slightly better, while SSE performs worse, especially with low local polynomial degree, because median splits do not target discontinuities well (Said et al., 16 Sep 2025).

Across these synthetic tests, the paper states that Tree-PCE achieves lower error for a given number of coefficients, handles discontinuities better than global PCE, often exceeds SSE because split choice is adaptive, and nevertheless requires balancing local degree and number of partitions.

The applied case study concerns a hydro-morphodynamic model of sediment transport in a curved channel. The challenge arises because bed elevation responds discontinuously to several uncertain parameters, especially the critical Shields number $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$6, the median sediment diameter $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$7, and the skin friction correction factor $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$8. The reported Tree-PCE configuration uses $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}} \Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$9, local degree $\mathcal{R}^r$0, an adaptive split grid of candidate thresholds, the sparse option, and up to 100 partitions. The global TSE is said to drop sharply and stabilize after about 27 significant partitions. The partitioning identifies regions of constant bed response versus regions of active erosion and deposition, and the paper states that the partitioning aligns with physically meaningful thresholds.

6. Accuracy–complexity trade-offs, limitations, and extensions

The paper emphasizes that Tree-PCE performance depends on a compromise among the number of subdomains, local polynomial degree, sparsity settings, candidate split grid size, stopping threshold $\mathcal{R}^r$1, and minimum node sample size $\mathcal{R}^r$2. Higher local degree improves local approximation, but it also increases the minimum sample size required per rectangle, which can reduce how much the tree can grow. This trade-off is central to both predictive accuracy and structural complexity (Said et al., 16 Sep 2025).

The stated advantages of Tree-PCE are that it handles nonlinear and discontinuous responses much better than global PCE, retains the analytical and interpretable nature of PCE locally, supports global sensitivity analysis, provides free variable-importance indices through the tree structure, can be combined with sparse regression for improved efficiency, and is fast to evaluate once the tree has been built. The stated limitations are that performance depends strongly on hyperparameters, local polynomial degree and number of subdomains must be balanced, analytical Sobol' formulas become costly when the tree is large, sufficient data are required in each subdomain or splitting is blocked, and for very complex high-dimensional problems the number of classes may still grow substantially.

The morphodynamic application also serves as the paper’s sensitivity-analysis demonstration. Tree-PCE Sobol' indices are reported to closely match pick-freeze reference results while using far fewer expensive model evaluations. The dominant parameters are ranked as $\mathcal{R}^r$3, then $\mathcal{R}^r$4, then $\mathcal{R}^r$5, followed by $\mathcal{R}^r$6 with mainly interaction effects. The parameters $\mathcal{R}^r$7, $\mathcal{R}^r$8, $\mathcal{R}^r$9, and $$G^{\mathrm{PCE}_{\mathcal{R}^r}}(\mathbf{x}) = \sum_{\boldsymbol{\alpha}\in\mathcal{A}^r} y^r_{\boldsymbol{\alpha}}\,\Psi^r_{\boldsymbol{\alpha}}(\mathbf{x}).$$0 are reported to have negligible influence. The Tree-PCE sensitivity indices reproduce the same ranking and add what the paper describes as a free structural interpretation of importance.

Several extensions are explicitly suggested: automated hyperparameter tuning, better split selection strategies, alternative local regression or sparsity methods, extension to larger-scale or more strongly dependent-input problems, and use of the Tree-PCE surrogate within Monte Carlo sensitivity workflows when analytic Sobol' computation becomes too expensive. The authors further suggest that tuning could be automated with tools such as Optuna using a Tree-structured Parzen Estimator (TPE) optimizer. This suggests that Tree-PCE is not only a fixed surrogate construction, but also a framework in which model architecture and estimation settings may be selected systematically to improve the accuracy–complexity compromise.