We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations. For boundary-compatible neural feature classes, the population error splits into approximation, empirical quadrature, and projected-gradient terms, with all non-approximation constants uniform in the microscopic scale \(\varepsilon\). Assuming a quantitative corrected \(H^1\)-estimate, a two-scale state class yields \[ \mathcal A_m^\varepsilon \le C\bigl(\varepsilon+Φ_{0,m_0}^2+Φ_{1,m_1}^2\bigr) \] in arbitrary dimension. We further introduce a convex primal-dual physics loss whose population value is a computable upper certificate for the state error. With additional flux-corrector regularity, a divergence-compatible two-scale flux class gives a certified state-flux bound combining \(O(\varepsilon)\) approximation, state and flux feature errors, empirical sampling error, and an \(O(K^{-1})\) optimization term. In contrast, for general periodic nonlinear fluxes satisfying a natural nondegeneracy condition, the empirical Rademacher complexities of strong-residual and squared-residual classes are bounded below by constant multiples of \((\varepsilon\sqrt N)^{-1}\) and \((\varepsilon^2\sqrt N)^{-1}\), respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted \(\varepsilon\)- and \(N\)-scalings for nonlinear fluxes in \(d=1,2,3\), validate every computed primal-dual certificate, and show that corrector-enriched classes substantially reduce energy and \(H^1\) errors as the microscopic scale is refined.
Multimodal Large Language Models (MLLMs) are often constrained by a language-space bottleneck, forcing complex visual reasoning into discrete tokens which can lose perceptual nuance. A promising alternative is continuous latent reasoning, where the goal is to discover implicit reasoning pathways that bridge the multimodal query and the final answer. However, this introduces a severe train-inference mismatch: a training-time posterior, conditioned on the ground-truth answer, can exploit answer-dependent shortcuts. Standard variational training then forces the inference-time prior to mimic a posterior that has access to information unavailable at test time, leading to poor performance. To address this, we propose Asymmetric Mutual Variational Learning (AMVL), a framework that resolves this mismatch via a bidirectional calibration objective. A forward KL divergence trains the target-agnostic prior to match the posterior, while a novel reverse KL divergence simultaneously regularizes the posterior, preventing it from collapsing into inference-incompatible regions and mitigating this ``answer leakage''. We provide theoretical analysis formalizing this leakage as prior contamination and prove that our dual-KL objective reduces it. We instantiate AMVL in a latent-integrated MLLM and show that it consistently outperforms strong discrete and latent-reasoning baselines, improving the average score on the complex BLINK benchmark by +10.83 and achieving gains of up to +32.00 on individual reasoning tasks, with analyses confirming improved latent-space stability.
Modern deep neural networks are trained using error backpropagation, which requires sequential forward and backward computations across network layers. As these networks become deeper, this introduces limitations, since layer-wise updates are strictly interdependent and cannot proceed in parallel. These constraints restrict training procedures to data-parallel schemes, thereby prohibiting model-parallel training. We propose TreeProp, an architecture-agnostic variational learning framework that organizes network layers into a tree-structured hierarchy. During training, TreeProp replaces sequential forward computations and backward gradient propagation with hierarchical computations. This allows intermediate representations and learning signals to be constructed in time complexity of $\mathcal{O}(\log N)$ for a network of $N$ layers. To the best of our knowledge, TreeProp is the first learning algorithm for deep neural networks with logarithmic parallel time complexity for both forward computation and backward gradient propagation during training. Furthermore, we show that multiple valid paths through the hierarchy exist, such that TreeProp implicitly learns subnetworks with different effective depths, but without additional training effort. We evaluate TreeProp on vision classification and autoregressive language modeling, matching the performance of conventional end-to-end training for a variety of tasks and outperforming previous contrastive training approaches. We further demonstrate the applicability of TreeProp to recurrent neural networks that otherwise rely on backpropagation through time.
Efficient quantum error correction is essential for the advancement of quantum computing. We propose a quantum neural network with a global structure that reduces the number of unitary matrices required in quantum circuits. This approach resulted in a 97% reduction in training time and up to a 25% improvement in the training completion rate, ultimately achieving a 100% success rate in training while surpassing the error correction performance reported in previous studies. In addition, we demonstrated the enhanced robustness of quantum error correction against internal network noise. Moreover, the fidelity of quantum error correction under internal network noise increased by up to 15% due to the reduced computational load.