CoDiQ-Generator: Unified Synthesis Framework

• CoDiQ-Generator is a unified framework that automates compiler-based DG code generation, controlled difficult question synthesis, and cooperative deep image modeling.
• It employs multi-stage pipelines including DSL parsing, intermediate representation optimization, and reinforcement learning to fine-tune performance across domains.
• Empirical evaluations demonstrate 6–10× speedups in DG code, increased normalized difficulty in generated questions, and state-of-the-art metrics in image synthesis.

CoDiQ-Generator is a term that refers to a set of research contributions across automated code generation for discontinuous Galerkin (DG) methods, test-time scaling for difficult question generation in large reasoning models (LRMs), and cooperative generative modeling. Each instance presents a distinct methodology but shares the underlying philosophy of controlled, efficient, and high-fidelity synthesis — whether of code, questions, or data samples. The following exposition focuses on three principal realizations of CoDiQ-Generator: (1) as a compiler-based system for DG methods, (2) as a neural generation framework for controllable difficult question generation, and (3) as a cooperative energy-based image generation algorithm.

1. Compiler-based Code Generation for Discontinuous Galerkin Methods

CoDiQ-Generator, in the context of finite element analysis, denotes a compiler-driven system for converting high-level DG variational forms into low-level, optimized C++ solver code. The system enables rapid prototyping and deployment of advanced DG schemes without manual kernel development. The user encodes mathematical forms in a Python-embedded domain-specific language (DSL) that closely mirrors published notation for trial/test spaces, integrals, and numerical fluxes. The pipeline includes five key stages: parsing the DSL, constructing an abstract syntax tree (AST), lowering forms to an intermediate representation (IR), performing symbolic and algebraic optimizations (CSE, precomputation, loop unrolling, vectorization), and emitting C++ kernel code.

A typical workflow is as follows: the researcher writes form definitions such as the Poisson interior-penalty DG bilinear in DSL syntax, compiles with CoDiQ-Generator to obtain C++ header/source files containing cell and facet kernel classes, and links these with solver drivers. The compiler statically precomputes reference tensors for each unique element and facet form, enabling run-time contractions to be emitted as straight-line code amenable to hardware autovectorization. Benchmarks demonstrate 6–10× speedups over hand-written code for problems ranging from Poisson to Stokes and advection–diffusion, with scalability persisting up to moderate polynomial degree and spatial dimension. The auto-generated code structure closely matches hand-optimized kernels, but with far lower development overhead. This facilitates rapid iteration and deployment of complex or bespoke DG schemes in research and industrial contexts (Ølgaard et al., 2011).

2. Test-Time Controllable Difficult Question Generation for LRMs

CoDiQ-Generator also refers to the controllable question synthesis framework described in "CoDiQ: Test-Time Scaling for Controllable Difficult Question Generation" (Peng et al., 2 Feb 2026). This system is based on further fine-tuning a Qwen3-8B LLM with an additional lightweight ValueNetwork (MLP) for difficulty assessment, reinforcement learning (RL) heads for feedback alignment, and an iterative pipeline for constructive hardening of seed questions. The generator operates via a test-time scaling mechanism, parameterized by a scaling factor $\alpha \in \mathbb{R}^+$ and a maximum rounds variable $T$, which together modulate the per-round reasoning token budget. Formally, at each round $i$, the system generates a new question $Q_i$, estimates its difficulty $d_i$ and solvability $s_i$ by

$$d = 1 - e^{-\kappa \alpha T}, \quad s = s_0 e^{-\lambda \alpha T},$$

where $\kappa$ and $\lambda$ are fitted, and $s_0 \approx 1$.

The system's pipeline prompts the LLM to iteratively "make $Q_{i-1}$ more difficult," tracks resulting $d_i$ and $s_i$, and applies RL to maximize the hardest solvable generated question while not exceeding a threshold (e.g., $s_i \geq 0.5$). User-exposed controls enable fine-grained tuning via $\alpha$, $T_{\mathrm{max}}$, and selectable hardening strategies (e.g., Dimensionality Constraints, CRT Abstraction). The ValueNetwork samples hidden states using a quadratic scheme over the first $4096$ tokens to extract K=10 snapshots for robust difficulty assessment. Generation employs nucleus sampling (temperature 0.7, top-p 0.9, top-k 50).

The principal innovation is in pushing the generator's "capacity ceiling"—the maximal combinatorial complexity of questions that can be produced while remaining verifiable. The hybrid verification pipeline (LLM ranking plus ValueNetwork plus Qwen3-32B solver) and RL boundary-failure mining extend the feasible (difficulty, solvability) frontier. Empirical evaluation on the CoDiQ-Bench demonstrates that CoDiQ-Gen-8B achieves a highest average normalized difficulty (DR-AVG 58.5%) among models tested and raises the upper difficulty ceiling to 62.4%, outperforming baseline and larger models. The CoDiQ-Corpus (44K items) generated via this approach significantly improves downstream reasoning on mathematical and programming benchmarks, yielding +7.5 point gains on AIME and consistent improvement on MATH-500. The system, corpus, and codebase are open-sourced to support further research (Peng et al., 2 Feb 2026).

3. Cooperative Descriptor–Generator Model for Deep Generative Modeling

CoDiQ-Generator also names the cooperative learning scheme introduced by Xie et al. for deep image modeling (Xie et al., 2016). Here, the architecture consists of two ConvNet-based models: an energy-based "descriptor" network $p_\theta(x)$ and a latent-variable "generator" $g_\alpha(z)$. The descriptor models $\exp[f_\theta(x)]/\mathcal{Z}(\theta)$ over a base density; the generator samples $x=g_\alpha(z)+\epsilon$, $z \sim \mathcal{N}(0, I)$.

Training alternates between these components via an "MCMC teaching" protocol. The generator produces ancestral samples $\tilde{x}_i$; the descriptor applies short-run Langevin dynamics to yield revised samples $x_i$, used for contrastive divergence updates of $\theta$. In turn, the generator learns to minimize the empirical residual $x_i - g_\alpha(z_i)$ over known latents to approach the revised samples, thus closing the MCMC loop. Formally, parameter updates are: $$\Delta \theta \propto \frac{1}{n}\sum \nabla_\theta f_\theta(x_{\mathrm{data}}) - \nabla_\theta f_\theta(x_{\mathrm{revised}}),$$

$$\Delta \alpha \propto \frac{1}{n\sigma^2} \sum [x_{\mathrm{revised}} - g_\alpha(z)] \nabla_\alpha g_\alpha(z).$$

This method unifies maximum likelihood learning of energy-based and generator models using finite-step Langevin revisions, producing higher-fidelity and more diverse unsupervised samples than conventional GAN, VAE, or contrastive divergence baselines. On image benchmarks (ImageNet, LSUN, CelebA, CIFAR-10), CoDiQ–Generator attains state-of-the-art inception scores and FID, with human perceptual studies confirming high sample quality. The approach offers robustness to mode collapse and supports dynamic texture synthesis when extended to spatial–temporal ConvNets (Xie et al., 2016).

4. Algorithmic and Pipeline Details

The three CoDiQ-Generator frameworks are unified by multi-stage pipelines structured around (i) high-level input grammars or prompts, (ii) intermediate lowering and optimization, (iii) domain-specific sampling or search, and (iv) feedback-coupled learning or code emission. In DG code generation (Ølgaard et al., 2011), the process is:

• Parse DSL forms → IR → algebraic/SIMD optimization → kernel code emission. In controllable question generation (Peng et al., 2 Feb 2026):
• Seed question → prompt engineering → iterative LLM/ValueNetwork/solvability pipeline → RL boundary updates and controlled difficulty progression. In cooperative generative modeling (Xie et al., 2016):
• Generator ancestral sampling → descriptor MCMC revision → feedback learning for both nets.

Direct pseudocode is available for each context:

• DG code: top-level AST/IR traversal and kernel contraction templates.
• Question generation: iterative RL and nucleus sampling, ValueNetwork state extraction, and solvability halting.
• Cooperative generation: alternated mini-batch loop over ancestral sampling, descriptor revision, parameter updates.

5. Quantitative Evaluation and Empirical Performance

The CoDiQ-Generator approaches provide quantitative outcomes exceeding hand-tuned or baseline alternatives. In DG code (Ølgaard et al., 2011), typical speedups reach 6–10× for element kernel evaluation, with assembly time scaling well in 2D and 3D across a range of $k$. In controllable question synthesis (Peng et al., 2 Feb 2026):

• CoDiQ-Gen-8B achieves DR-AVG 58.5% (CoDiQ-Bench), outperforms Qwen3-32B, and sets new upper difficulty bounds.
• CoDiQ-Corpus yields 87.1% average normalized difficulty in sampled competition-grade questions (vs. AIME 51.5%, LiveCodeBench 42.3%) with human-verified solvability >82%.
• Downstream training leads to +2% on MATH-500, +7.5% on AIME for curriculum RL-trained LRMs.

The cooperative descriptor–generator approach (Xie et al., 2016) demonstrates superior IS, FID, and sample log-likelihood relative to DCGAN, W-GAN, VAE, and separate learning, with a qualitative absence of mode collapse and improved sample realism in both static and dynamic image domains.

6. Design Implications and Significance

The CoDiQ-Generator concept, in all realizations, yields new upper bounds for automatically synthesized domain content — question difficulty, code efficiency, or data sample realism. By providing fine-grained user control (via prompts, DSL, or algorithmic scaling parameters), and incorporating feedback granularity (difficulty/solvability networks, symbolic IR optimization, energy-based sampling), these frameworks enable not only superior performance metrics but also improved reproducibility and extensibility for downstream research and engineering tasks. An important implication is that such architectures facilitate rapid exploration and optimization of complex, structured synthesis in scientific and machine learning workloads, supporting both automated curriculum construction and high-performance code deployment.

7. Future Directions and Research Outlook

Continued research into CoDiQ-Generator frameworks is likely to focus on extending their scaling, generality, and integration. In DG code synthesis, directions include the support of more general PDE systems and exotic element bases. For controllable question generation, further advances may be anticipated in open-ended curriculum design, transfer to new reasoning domains, and robustness at higher token budgets or reasoning depths. Cooperative generative modeling admits further extension to multimodal and sequential synthesis, as well as theoretical analysis of convergence guarantees and sample coverage. In all domains, the emphasis on test-time controllability, feedback-driven optimization, and high-level input abstraction is poised to set new standards for scalable, reliable automated synthesis in both scientific computing and deep learning (Ølgaard et al., 2011, Xie et al., 2016, Peng et al., 2 Feb 2026).