In this work, we benchmark \simulacra's synthetic data generation pipeline against a state-of-the-art Microsoft pipeline on a dataset of small to large systems. By analyzing the energy quality, autocorrelation times, and effective sample size, our findings show that Simulacra's Large Wavefunction Models (LWM) pipeline, paired with state-of-the-art Variational Monte Carlo (VMC) sampling algorithms, reduces data generation costs by 15-50x, while maintaining parity in energy accuracy, and 2-3x compared to traditional CCSD methods on the scale of amino acids. This enables the creation of affordable, large-scale \textit{ab-initio} datasets, accelerating AI-driven optimization and discovery in the pharmaceutical industry and beyond. Our improvements are based on a novel and proprietary sampling scheme called Replica Exchange with Langevin Adaptive eXploration (RELAX).

We propose a novel framework for learning hidden graph structures from data using geometric analysis and nonlinear dynamics. Our approach: (1) Defines discrete Sobolev spaces on graphs for scalar/vector fields, establishing key functional properties; (2) Introduces gauge-equivalent nonlinear Schrödinger and Landau--Lifshitz dynamics with provable stable stationary solutions smoothly dependent on input data and graph weights; (3) Develops a stochastic gradient algorithm over graph moduli spaces with sparsity regularization. Theoretically, we guarantee: topological correctness (homology recovery), metric convergence (Gromov--Hausdorff), and efficient search space utilization. Our dynamics-based model achieves stronger generalization bounds than standard neural networks, with complexity dependent on the data manifold's topology.

In the previous article, we presented a quantum-inspired framework for modeling semantic representation and processing in Large Language Models (LLMs), drawing upon mathematical tools and conceptual analogies from quantum mechanics to offer a new perspective on these complex systems. In this paper, we clarify the core assumptions of this model, providing a detailed exposition of six key principles that govern semantic representation, interaction, and dynamics within LLMs. The goal is to justify that a quantum-inspired framework is a valid approach to studying semantic spaces. This framework offers valuable insights into their information processing and response generation, and we further discuss the potential of leveraging quantum computing to develop significantly more powerful and efficient LLMs based on these principles.

We propose a novel clustering method that is an extension of ideas inher- ent to scale-space clustering and support-vector clustering. Like the lat- ter, it associates every data point with a vector in Hilbert space, and like the former it puts emphasis on their total sum, that is equal to the scale- space probability function. The novelty of our approach is the study of an operator in Hilbert space, represented by the Schr odinger equation of which the probability function is a solution. This Schr odinger equation contains a potential function that can be derived analytically from the probability function. The method has one variable parameter, the scale of its Gaussian kernel.

Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China (Dated: March 12, 2018) Motivated by the question that whether the empirical fitting of data by neural networks can yield the same structure of physical laws, we apply neural networks to a quantum mechanical two-body scattering problem with short-range potentials--a problem by itself plays an important role in many branches of physics. After training, the neural network can accurately predict s - wave scattering length, which governs the low-energy scattering physics. By visualizing the neural network, we show that it develops perturbation theory order by order when the potential depth increases, without solving the Schr odinger equation or obtaining the wavefunction explicitly. The result provides an important benchmark to the machine-assisted physics research or even automated machine learning physics laws. Human physicists have made great achievements in discovering laws of physics during the last several centuries.

While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from {\em small} data. In particular, we introduce \emph{hidden physics models}, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier-Stokes, Schr\"odinger, Kuramoto-Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data.