Efficient Deep Learning Methods for Solving High-dimensional Partial Differential Equations for Applications in Option Pricing
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Bibliographic record
Abstract
Partial Differential Equations (PDEs) are used to model a variety of dynamical systems around us. Recent advances in deep learning have enabled us to solve these PDEs in higher dimensions by addressing the Curse of Dimensionality (COD). However, these approaches are constrained by training time and memory. To tackle these shortcomings, we introduce three approaches starting with Multi-Level Dense Neural Networks (ML-DNN). ML-DNN draws inspiration from Multi-Level Monte-Carlo to efficiently sample and perform hierarchical learning thereby providing substantial time savings compared to the classical Dense Neural Network (DNN). Next, we implement Tensor Neural Networks, a quantum-inspired architecture that provides significant parameter savings and faster convergence while attaining the same accuracy as compared to a DNN. Finally, we introduce a model-based Reinforcement Learning algorithm which addresses the COD and is independent of the PDE family. We benchmark these models on parabolic PDEs, empirically showing their advantages over the current state-of-the-art models.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it