Deep Semi-Parametric Models (SPMs)

• Deep Semi-Parametric Models (SPMs) are models that integrate a structured parametric core with flexible nonparametric components, balancing interpretability and adaptability.
• They are applied across diverse domains including survival analysis, psychophysics, PDE-governed dynamics, continual learning, and machine unlearning, thereby retaining domain-specific inductive bias.
• Key design strategies include constrained deep components, expert mixtures, memory-based replay, and profiling techniques that enable both accurate prediction and reliable inference.

Deep Semi-Parametric Models (SPMs) are models that combine a parametric component with a flexible component learned from data, while preserving explicit structural commitments that would be absent in a purely black-box formulation. Across the literature represented here, the parametric component serves different roles: an unspecified baseline hazard with a structured risk function in survival analysis, a psychometric curve along a designated intensity axis in sensory discrimination, a physics-informed operator inside a partial differential equation, a parametric recognizer coupled to a semi-parametric memory in continual learning, or a standard deep backbone augmented with a non-parametric test-time set in machine unlearning. The resulting family is not unified by a single architecture, but by a recurring design principle: retain an interpretable or otherwise constrained core, and attach a nonparametric or deep mechanism only where flexibility is needed (Nagpal et al., 2019, Keeley et al., 2023, Chen et al., 28 Jun 2025, Liu et al., 20 Apr 2025, Zheng et al., 24 Mar 2026).

1. Conceptual definition and recurring design pattern

In the supplied literature, semiparametric structure consistently means that only part of the model is fully parameterized. In survival analysis, the canonical example is Cox Proportional Hazards (CPH), where the hazard is written as

$h(t|x) = h_0(t)\exp(\eta(x)),$

with the baseline hazard $h_0(t)$ left unspecified and the covariate-dependent log-risk $\eta(x)$ modeled either linearly or through a more flexible function (Nagpal et al., 2019). In psychophysics, the semiparametric decomposition makes stimulus intensity “special,” preserving a traditional psychometric form along that dimension while allowing the psychometric parameters to vary nonparametrically over contextual dimensions via Gaussian process (GP) priors (Keeley et al., 2023). In PDE-governed systems, the semiparametric split places known physical mechanisms in a parametric operator and unknown effects in a neural nuisance term (Chen et al., 28 Jun 2025).

A common feature of deep SPMs is that the flexible component is not arbitrary; it is constrained by the role assigned to it. In the sensory discrimination model, the GP governs threshold and slope variation across context rather than replacing the psychometric form itself (Keeley et al., 2023). In SemiPDE, the neural network models the unknown operator $\mathcal{F}$ while the PDE solver enforces the governing equation, boundary conditions, and initial conditions (Chen et al., 28 Jun 2025). In unlearning, the non-parametric component is a set of per-sample embeddings fused into the forward pass, so that deletion of selected samples changes predictions without editing model parameters (Zheng et al., 24 Mar 2026). This suggests that “deep” in deep SPMs does not imply abandonment of structure; rather, it denotes a structured allocation of expressivity.

The literature also shows that semiparametricity serves different technical goals. In some cases it is primarily statistical, as with unspecified baseline hazards or orthogonal nuisance modeling. In others it is operational, as in continual learning or unlearning, where semi-parametric memory externalizes part of the learned information instead of fully absorbing it into weights (Liu et al., 20 Apr 2025, Zheng et al., 24 Mar 2026). A plausible implication is that deep SPMs are best understood as a design paradigm rather than a narrow model class.

2. Survival analysis: Cox-type deep parameterizations and interpretable mixtures

The survival-analysis line of work defines deep SPMs by preserving the semi-parametric Cox setup while replacing the linear risk score with a nonlinear function. In linear CPH,

$\eta(x)=x^\top \beta,$

and the partial log-likelihood is

$$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$

where $R_i:=\{j:t_j\ge t_i\}$ is the risk set (Nagpal et al., 2019). DeepSurv-style models keep the unspecified baseline hazard and substitute

$\eta(x)=f_\theta(x),$

with $f_\theta$ an MLP trained by maximizing the Cox partial likelihood, often with regularization

$$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$

The experimental comparisons noted standard choices such as MLPs with ReLU, SELU, or Sigmoid, Adam or RMSprop, and $h_0(t)$0 regularization (Nagpal et al., 2019).

The principal alternative proposed in this literature is Cox Mixture of Experts (MoCE), an interpretable semi-parametric model built from $h_0(t)$1 linear Cox experts and a gating function. The gating weights are

$h_0(t)$2

and the nonlinear semi-parametric risk is

$h_0(t)$3

with $h_0(t)$4 for linear experts. The paper distinguishes soft-gating inference,

$h_0(t)$5

from hard-gating inference,

$h_0(t)$6

Because exact marginalization over latent expert assignments has complexity $h_0(t)$7, the paper adopts Rosen’s approximation and then proposes an ELBO objective with complexity $h_0(t)$8 (Nagpal et al., 2019).

The ELBO is

$h_0(t)$9

and parameters are learned by maximizing the sum of ELBOs over observed events with $\eta(x)$0 regularization on the experts (Nagpal et al., 2019). The paper states that the ELBO is a lower bound on the log-likelihood and also a lower bound on Rosen’s approximation, via Jensen’s inequality.

Empirically, the reported comparisons use METABRIC, SUPPORT, and Rotterdam-GBSG, with concordance index (C-index) and bootstrapped confidence intervals. For linear gating, ELBO-MoCE-SG achieved $\eta(x)$1 on METABRIC, compared with DeepSurv at $\eta(x)$2, and $\eta(x)$3 on SUPPORT, compared with DeepSurv at $\eta(x)$4; on Rotterdam-GBSG, ELBO-MoCE-HG achieved $\eta(x)$5, compared with DeepSurv at $\eta(x)$6 (Nagpal et al., 2019). With nonlinear gating, ELBO-MoCE-SG reached $\eta(x)$7 on METABRIC and $\eta(x)$8 on SUPPORT. The paper further reports that ELBO optimization outperforms Rosen’s RT-MoCE throughout training and is robust to overfitting compared to RT-MoCE.

A complementary experimental study compared Cox proportional hazards, Aalen’s additive regression model, Weibull AFT, Random Survival Forest, Gradient Boosting with Cox Proportional Hazards Loss, and DeepSurv on PBC and GBCSG2 through the C-index (Fernandez et al., 2020). In that study, DeepSurv did not outperform the best tree-based methods on either dataset. On PBC, tuned RSF achieved the highest C-index distribution, though its advantage over tuned Cox was reported as less than $\eta(x)$9 in C-index. On GBCSG2, RSF+ANN attained the best C-index distribution, with an improvement of approximately $\mathcal{F}$0 over tuned Cox (Fernandez et al., 2020). Taken together, these papers oppose the simple equation of “deep” with “better”: the survival literature here argues that semiparametric structure is often better exploited through interpretable localized linearity than through unrestricted overparameterization.

3. High-dimensional psychophysics: parametric psychometrics with nonparametric context dependence

In sensory discrimination, the semiparametric construction begins from the classical psychometric function

$\mathcal{F}$1

where $\mathcal{F}$2 is an intensity dimension and $\mathcal{F}$3 denotes contextual dimensions (Keeley et al., 2023). The core idea is to preserve the psychometric form along $\mathcal{F}$4 while placing GP priors on psychometric parameters as functions of $\mathcal{F}$5. In the slope-offset parameterization used in the paper,

$\mathcal{F}$6

with

$\mathcal{F}$7

The slope GP mean is centered at a positive constant, with $\mathcal{F}$8 in all experiments, enforcing a prior preference for positive monotonicity in intensity (Keeley et al., 2023).

The paper gives a canonical latent construction with

$\mathcal{F}$9

and

$\eta(x)=x^\top \beta,$0

where “$\eta(x)=x^\top \beta,$1” denotes the Hadamard product. The binary observation model is then $\eta(x)=x^\top \beta,$2. For inference, the paper uses factorized variational inference with ELBO

$\eta(x)=x^\top \beta,$3

with the expectation approximated using $\eta(x)=x^\top \beta,$4D Gauss–Hermite quadrature per datapoint (Keeley et al., 2023).

A second approximation replaces the latent Hadamard product by a multivariate normal:

$\eta(x)=x^\top \beta,$5

which induces the kernel

$\eta(x)=x^\top \beta,$6

This MVN Hadamard approximation reduces variational parameters and supports analytic acquisition functions that require MVN posteriors (Keeley et al., 2023).

The semiparametric decomposition is especially consequential for active learning. For a target response level $\eta(x)=x^\top \beta,$7, the threshold conditioned on context is

$\eta(x)=x^\top \beta,$8

and the paper proposes ThresholdBALV,

$\eta(x)=x^\top \beta,$9

then sets $$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$0 at the posterior mean of $$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$1 (Keeley et al., 2023). The acquisition directly targets threshold uncertainty rather than generic response uncertainty.

Empirically, on synthetic 2D and 8D tasks, SPM variants achieved lower Expected Brier score with far fewer samples than GP-RBF and GP-linear baselines, while all models converged with sufficient data (Keeley et al., 2023). On real visual psychophysics datasets, using 15-fold CV and mean held-out log-likelihood, SPM variants consistently yielded superior held-out log-likelihood over GP-RBF and GP-linear across participants. In active learning on the 8D synthetic setting, ThresholdBALV with SPM performed strongly early in acquisition, while GlobalMI + MVN SPM and GP-RBF with GlobalMI could be competitive by the end. The paper also reports that quasi-random Sobol sampling is surprisingly strong.

The broader relevance to deep SPMs is explicit in the paper’s extensions: deep kernel learning, Bayesian neural parameterizations of $$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$2, and Neural Processes or Deep Gaussian Processes for the context-dependent psychometric parameters (Keeley et al., 2023). This suggests a template in which the semiparametric core encodes domain-specific inductive bias and the deep component handles high-dimensional context without discarding that bias.

4. PDE-governed dynamics: profiling, orthogonality, and inference with deep nuisance components

The PDE literature presents perhaps the most explicit general theory of deep semiparametric modeling. SemiPDE is formulated as

$$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$3

with observation model

$$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$4

where the parametric part is the operator $$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$5 and the nonparametric part is the unknown differential operator $$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$6, modeled by a neural network (Chen et al., 28 Jun 2025). The same framework is illustrated with reaction–diffusion and Navier–Stokes examples, including

$$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$7

and

$$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$8

The estimation strategy is deep profiling M-estimation. For PDE solution $$\ell(\theta)=\sum_{i:e_i=1}\Big(\eta(x_i)-\log \sum_{j\in R_i}\exp(\eta(x_j))\Big),$$9 obtained numerically for given $R_i:=\{j:t_j\ge t_i\}$0, the empirical criterion is

$R_i:=\{j:t_j\ge t_i\}$1

with squared loss $R_i:=\{j:t_j\ge t_i\}$2 and penalty $R_i:=\{j:t_j\ge t_i\}$3 (Chen et al., 28 Jun 2025). The profiled criterion

$R_i:=\{j:t_j\ge t_i\}$4

decouples the solution of the PDE from the statistical estimation step. The paper contrasts this directly with PINNs: instead of parameterizing the solution $R_i:=\{j:t_j\ge t_i\}$5 itself with a neural network and enforcing the PDE through residual penalties, SemiPDE uses numerical PDE solvers to compute $R_i:=\{j:t_j\ge t_i\}$6 and profiles out the nuisance.

The algorithm iterates between numerical solution, gradient computation, and parameter updates. Gradients are

$R_i:=\{j:t_j\ge t_i\}$7

$R_i:=\{j:t_j\ge t_i\}$8

with sensitivities computed by the adjoint method and automatic differentiation through the neural nuisance term (Chen et al., 28 Jun 2025).

The paper’s main theoretical claims concern rates and inference. Under Assumptions 1–4, the nuisance estimator obeys

$R_i:=\{j:t_j\ge t_i\}$9

and the solution estimator satisfies

$\eta(x)=f_\theta(x),$0

Under Neyman orthogonality and an efficient orthogonal direction, the parametric estimator is asymptotically normal,

$\eta(x)=f_\theta(x),$1

and efficient under Assumption 6, with

$\eta(x)=f_\theta(x),$2

The paper further gives a finite-difference variance estimator and confidence intervals with coverage

$\eta(x)=f_\theta(x),$3

(Chen et al., 28 Jun 2025).

Empirically, the paper reports four PDE simulation cases spanning reaction–diffusion, multi-species systems, Nernst–Planck, and 3D Navier–Stokes. SemiPDE achieved lower errors in estimating both $\eta(x)=f_\theta(x),$4 and $\eta(x)=f_\theta(x),$5 across sample sizes and noise levels than parametric PDE regression, nonparametric regression with deep nets, and PINN-style joint training. Repeated experiments showed small bias, accurate variance estimation, and coverage rates matching nominal levels. In real applications, SemiPDE achieved lower test MSE than Fisher–Kolmogorov and Porous–Fisher baselines in in vitro cell culture assays, and estimated vegetation drag coefficients $\eta(x)=f_\theta(x),$6 more stably and accurately than parametric PDE-only and direct force-based baselines in wave propagation through vegetation (Chen et al., 28 Jun 2025).

This body of work treats deep SPMs not merely as a modeling convenience but as an inferential regime. A plausible implication is that semiparametricity becomes especially valuable when the objective includes both prediction and valid inference for interpretable parameters under complex nuisance structure.

5. Semi-parametric memory in continual learning

In continual learning, the semiparametric split moves from statistical form to memory architecture. BrainCL consists of a task-agnostic recognition network

$\eta(x)=f_\theta(x),$7

with prediction

$\eta(x)=f_\theta(x),$8

together with a non-parametric memory

$\eta(x)=f_\theta(x),$9

where $f_\theta$0 is an arithmetic-encoded bitstream of a quantized memory cue, and a set of task-dependent pattern completion networks $f_\theta$1 reconstruct stored cues during replay (Liu et al., 20 Apr 2025). Unlike retrieval-augmented inference models, the memory is not queried at test time to produce predictions; it supplies replay data during consolidation.

For a sample $f_\theta$2 of task $f_\theta$3, pattern separation produces $f_\theta$4 with $f_\theta$5, where $f_\theta$6 and spatial dimensions are reduced by a factor of $f_\theta$7. Quantization yields $f_\theta$8, while differentiable training uses $f_\theta$9 with $$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$0. Arithmetic coding stores $$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$1, and decoding followed by task-specific completion gives $$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$2 (Liu et al., 20 Apr 2025).

The memory module optimizes two objectives. Recall fidelity is

$$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$3

and entropy modeling is

$$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$4

with $$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$5 minimizing

$$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$6

where $$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$7 (Liu et al., 20 Apr 2025). The wake–sleep schedule is deterministic per task. During wake, the memory module is trained for $$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$8 epochs with Adam, then $$\mathcal{L}(\theta)=-\ell(\theta)+\lambda \Omega(\theta).$$9 is frozen and only $h_0(t)$00 is trained on recalled samples from working memory for $h_0(t)$01 epochs with SGD. During sleep, a class-conditional representativeness rule selects low-distance samples per class, based on

$h_0(t)$02

to transfer to long-term memory, after which the full model $h_0(t)$03 is finetuned on class-balanced replay for $h_0(t)$04 epochs (Liu et al., 20 Apr 2025).

The paper emphasizes that there are no EWC-like quadratic regularizers or knowledge-distillation terms. Stability comes from freezing $h_0(t)$05 during wake and replay from long-term memory during sleep. The read operation reconstructs replay samples by decoding stored bitstreams and applying the appropriate task-specific decoder:

$h_0(t)$06

Only the cues $h_0(t)$07 and decoders $h_0(t)$08 persist after working memory is cleared (Liu et al., 20 Apr 2025).

Empirically, BrainCL is evaluated on class-incremental learning on ImageNet-100, CoRe50, and MedMNIST. On ImageNet-100, BrainCL with memory size $h_0(t)$09 MB achieved $h_0(t)$10 and $h_0(t)$11, compared with Replay at $h_0(t)$12, $h_0(t)$13, iCaRL at $h_0(t)$14, $h_0(t)$15, and DER at $h_0(t)$16, $h_0(t)$17 (Liu et al., 20 Apr 2025). On CoRe50, BrainCL achieved $h_0(t)$18, $h_0(t)$19 with $h_0(t)$20 MB, and on MedMNIST it achieved $h_0(t)$21, $h_0(t)$22 with $h_0(t)$23 MB. The paper also reports improved robustness under ImageNet-C corruptions, stronger performance on long-tailed ImageNet-100-LT, higher AUROC on ImageNet-O, preservation of feature diversity close to joint training, and reduced class-wise disparate impact across tasks (Liu et al., 20 Apr 2025).

This formulation broadens the scope of deep SPMs. The non-parametric component is not a kernel smoother or an external retrieval head used at deployment, but an episodic store that shapes parameter updates during an offline consolidation phase. That shift in where semiparametric interaction occurs is one of the clearest conceptual departures in the supplied literature.

6. Deletion-ready deep SPMs for machine unlearning

The unlearning literature defines deep SPMs by their explicit dependence on a set of training-sample embeddings at test time. Instead of a purely parametric prediction

$h_0(t)$24

an SPM predicts

$h_0(t)$25

and can be unlearned by evaluating

$h_0(t)$26

for a forget set $h_0(t)$27, without updating parameters (Zheng et al., 24 Mar 2026). The architecture contains a parametric branch $h_0(t)$28, a non-parametric branch $h_0(t)$29 that encodes each training point into an instance embedding, and a fusion module $h_0(t)$30 that aggregates the set with the current latent state. The layerwise recursion is

$h_0(t)$31

For classification, the fusion module aggregates instance embeddings except the self-instance:

$h_0(t)$32

and the final prediction is

$h_0(t)$33

with attention weights

$h_0(t)$34

For diffusion generation, the paper defines a patch-aligned fusion in the UNet mid-block using Bahdanau-style attention (Zheng et al., 24 Mar 2026).

The central operational claim is that unlearning becomes a data operation. If per-sample embeddings are stored, deletion removes entries from the index and from the set $h_0(t)$35; if class prototypes are used, deletion updates the prototypes to exclude forgotten samples. The paper reports deletion time $h_0(t)$36 and measured unlearning time under $h_0(t)$37 second in all experiments (Zheng et al., 24 Mar 2026). It also introduces two variants for scalability: SPM-R, which retrieves top-$h_0(t)$38 neighbors through FAISS with PQ codes and uses $h_0(t)$39, and SPM-C, which averages embeddings per class to obtain a constant-sized set (Zheng et al., 24 Mar 2026).

The paper evaluates classification performance on CIFAR-10 and ImageNet-1K. On CIFAR-10, ResNet18 achieved $h_0(t)$40 top-1, ResNet18-KNN achieved $h_0(t)$41, SPM-C achieved $h_0(t)$42, and SPM-R achieved $h_0(t)$43 (Zheng et al., 24 Mar 2026). On ImageNet-1K, ResNet18-KNN achieved $h_0(t)$44, SPM-C achieved $h_0(t)$45, and SPM-R achieved $h_0(t)$46. The paper states that using a larger inputted set at test time improves accuracy for both KNN and SPMs, and that SPM-C halves runtime relative to KNN on ImageNet, $h_0(t)$47s versus $h_0(t)$48s (Zheng et al., 24 Mar 2026).

For unlearning on CIFAR-10 classification, one-class deletion with SPM achieved $h_0(t)$49, $h_0(t)$50, $h_0(t)$51, $h_0(t)$52, $h_0(t)$53, and $h_0(t)$54; five-class deletion yielded $h_0(t)$55, $h_0(t)$56, $h_0(t)$57, $h_0(t)$58, $h_0(t)$59, and $h_0(t)$60 (Zheng et al., 24 Mar 2026). On ImageNet-1K one-class unlearning, SPM deletion achieved $h_0(t)$61, $h_0(t)$62, $h_0(t)$63, $h_0(t)$64, $h_0(t)$65, with $h_0(t)$66, while fine-tuning and SalUn required hours (Zheng et al., 24 Mar 2026). The abstract states that on ImageNet classification, SPMs reduce the prediction gap relative to a retrained oracle baseline by $h_0(t)$67 and achieve over $h_0(t)$68 faster unlearning than existing approaches on parametric models.

The paper also gives a theoretical relation among evaluation criteria. If the oracle margin satisfies $h_0(t)$69 for all $h_0(t)$70, then

$h_0(t)$71

This establishes $h_0(t)$72 and especially $h_0(t)$73 as stricter than accuracy-gap comparisons (Zheng et al., 24 Mar 2026).

Within the broader SPM landscape, this work is unusual because the non-parametric branch is designed specifically to support forgetting. The paper’s phrase “non-parametric behavior during unlearning” means that outputs depend on a set of per-sample embeddings during the forward pass, so deletion immediately changes the prediction map. This is a stronger operational notion than the usual semiparametric claim of retaining interpretability.

7. Comparative themes, misconceptions, and domain-specific trade-offs

A recurring misconception is that deep SPMs are simply parametric models with an auxiliary deep module attached. The supplied literature instead presents semiparametric structure as the principal modeling decision and the deep component as subordinate to that structure. In survival analysis, replacing $h_0(t)$74 by $h_0(t)$75 keeps the unspecified baseline hazard and Cox partial likelihood intact, so DeepSurv remains semi-parametric in time (Fernandez et al., 2020). In psychophysics, the deep or GP component governs variation of interpretable psychometric parameters over context rather than replacing the psychometric law (Keeley et al., 2023). In SemiPDE, the neural nuisance does not replace the PDE solver; it enters an operator decomposition whose estimation is profiled and whose parametric part remains inferentially accessible (Chen et al., 28 Jun 2025).

Another misconception is that deep SPMs always improve predictive performance relative to classical or purely machine-learning alternatives. The survival papers are explicit that this is not guaranteed. “Nonlinear Semi-Parametric Models for Survival Analysis” argues against overly parameterized deep models and reports that more interpretable semiparametric models inspired by mixtures of experts perform equally well or better in several settings (Nagpal et al., 2019). The experimental comparison on PBC and GBCSG2 similarly reports that DeepSurv did not outperform the best tree-based methods (Fernandez et al., 2020). By contrast, the psychophysics, PDE, continual learning, and unlearning papers present concrete settings in which the semiparametric design yields strong empirical gains, but those gains arise from preserving the right inductive bias, not from depth alone (Keeley et al., 2023, Chen et al., 28 Jun 2025, Liu et al., 20 Apr 2025, Zheng et al., 24 Mar 2026).

The literature also reveals several distinct trade-offs. In survival analysis, linear gating improves interpretability, while nonlinear gating can improve C-index at some interpretability cost (Nagpal et al., 2019). In psychophysics, the exact SPM benefits from task-specific floors and flexible links, whereas the MVN approximation is faster and more compatible with analytic acquisition functions but can oversmooth when floors are imposed (Keeley et al., 2023). In SemiPDE, profiling and solver accuracy improve stability and support semiparametric inference, but repeated PDE solves and adjoint sensitivities impose nontrivial computational cost (Chen et al., 28 Jun 2025). In BrainCL, inference-time cost is identical to a standard ResNet-18 classifier because memory is used only during replay, but the number of decoders grows with tasks (Liu et al., 20 Apr 2025). In unlearning SPMs, deletion is immediate, yet attention over large inputted sets creates a trade-off among memory, runtime, and performance; retrieval and clustering mitigate but do not eliminate this cost (Zheng et al., 24 Mar 2026).

Across domains, interpretability takes different forms. In MoCE, it is coefficient-level interpretability of localized linear Cox experts and, with linear gating, transparent region assignment (Nagpal et al., 2019). In psychophysics, it is threshold, slope, guess rate, and lapse rate, together with monotonicity along intensity (Keeley et al., 2023). In SemiPDE, it is inference on physical coefficients and identification of unknown mechanisms through orthogonal nuisance modeling (Chen et al., 28 Jun 2025). In BrainCL and unlearning SPMs, interpretability is less coefficient-centric; instead, it lies in explicit memory operations, replay pathways, and deletion mechanisms (Liu et al., 20 Apr 2025, Zheng et al., 24 Mar 2026).

The supplied papers therefore depict deep SPMs as a heterogeneous but coherent research area. Their common claim is not that one architecture solves every problem, but that carefully chosen semiparametric structure can preserve inductive bias, enable stronger inference or memory control, and in some settings outperform both classical parametric models and unconstrained deep alternatives. A plausible implication is that future work on deep SPMs will continue to be driven less by generic architectural scaling than by domain-specific choices about which components should remain structured, which should remain externalized, and which should be learned flexibly.