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Abstract Simulating quantum systems in a finite volume is a powerful theoretical tool to extract information about them. Real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume. This approach is relevant not only for nuclear physics, where lattice methods for few- and many-nucleon states complement phenomenological shell-model descriptions and ab initio calculations of atomic nuclei based on harmonic oscillator expansions, but also for other fields such as simulations of cold atomic systems. This contribution presents recent progress concerning finite-volume simulations of few-body systems. In particular, it discusses details regarding the efficient numerical implementation of separable interactions and it presents eigenvector continuation as a method for performing robust and efficient volume extrapolations.
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How Do Transformers Model Physics? Investigating the Simple Harmonic Oscillator
How do transformers model physics? Do transformers model systems with interpretable analytical solutions or do they create an “alien physics” that is difficult for humans to decipher? We have taken a step towards demystifying this larger puzzle by investigating the simple harmonic oscillator (SHO), x¨+2γx˙+ω02x=0, one of the most fundamental systems in physics. Our goal was to identify the methods transformers use to model the SHO, and to do so we hypothesized and evaluated possible methods by analyzing the encoding of these methods’ intermediates. We developed four criteria for the use of a method within the simple test bed of linear regression, where our method was y=wx and our intermediate was w: (1) Can the intermediate be predicted from hidden states? (2) Is the intermediate’s encoding quality correlated with the model performance? (3) Can the majority of variance in hidden states be explained by the intermediate? (4) Can we intervene on hidden states to produce predictable outcomes? Armed with these two correlational (1,2), weak causal (3), and strong causal (4) criteria, we determined that transformers use known numerical methods to model the trajectories of the simple harmonic oscillator, specifically, the matrix exponential method. Our analysis framework can conveniently extend to high-dimensional linear systems and nonlinear systems, which we hope will help reveal the “world model” hidden in transformers.
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- Award ID(s):
- 2019786
- PAR ID:
- 10570066
- Publisher / Repository:
- MDPI
- Date Published:
- Journal Name:
- Entropy
- Volume:
- 26
- Issue:
- 11
- ISSN:
- 1099-4300
- Page Range / eLocation ID:
- 997
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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