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Energy Manifold Natural Gradient Descent: From Riemannian Optimization to Modern Neuroscience, NeuroAI and Quantum Physics

Dr. Rishabh Pathak
When Geometry, Energy, Artificial Intelligence and Neuroscience Converge Modern Artificial Intelligence is rapidly moving beyond the idea that learning simply means minimizing an error function. Increasingly, researchers are asking a deeper question: what is the structure of the space in which learning takes place? This question becomes particularly important when the system being modelled is constrained, nonlinear, dynamic, or governed by physical principles. A recent work titled “Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers” , by Zhangyong Liang and Huanhuan Gao, introduces Energy Manifold Natural Gradient Descent (EMNGD) , a mathematical framework that extends energy-based natural-gradient optimization from unconstrained Euclidean parameter spaces to constrained Riemannian parameter manifolds . At its core, the framework proposes a simple but powerful principle: An optimization algorithm should not only determine how to reduce error; it sh...
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Problems Machine Learning Can Solve.

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Dr. Rishabh Pathak


1. What Problems Can Machine Learning Solve?

Machine learning is particularly effective for automating decision-making by generalizing from data examples. The core strength of machine learning lies in its ability to learn from input/output pairs and then apply learned knowledge to new, unseen data.

2. Supervised Learning Problems

  • Definition: Supervised learning refers to tasks where the algorithm is trained on labeled data — input data where the desired output or target is known.
  • How it Works: A user provides the model with many examples (input/output pairs). The model learns the mapping from inputs to outputs.
  • Prediction Goal: The goal is to make accurate predictions on new inputs whose outputs are unknown.

Example Use Cases:

·         Spam Detection: The input is email features; the output is a label indicating spam or not spam. The system learns from many labeled emails and predicts the label on new emails.

·         Handwritten Digit Recognition: The input is images of handwritten digits, the output is the true digit label. The system learns from scanned envelopes with labeled digits.

·         Fraud Detection: Input data includes user transaction details, while the output is whether a transaction is fraudulent. Fraud labels come from customer reports over time.

Why Suitable:

·         Supervised learning excels when you can collect supervised datasets.

·         It automates tasks that would be time-consuming or costly to do manually.

·         It’s easy to evaluate performance using objective metrics since labeled data is available.

3. Unsupervised Learning Problems

  • Definition: Unsupervised learning is used when only input data is available without corresponding labels.
  • Purpose: It seeks to find hidden structure, patterns, or themes within the data.

Example Use Cases:

·         Topic Modeling: Given a large collection of blog posts (text data), unsupervised algorithms can identify underlying themes or topics without predefined labels.

Challenges:

·         Results can be more difficult to interpret.

·         The absence of labeled outputs makes it harder to measure success precisely.

4. General Criteria for Applying Machine Learning

Before applying machine learning algorithms, one should consider:

  • Is the data representative and sufficient to capture the problem?
  • Can the problem be phrased as a prediction from given inputs to outputs?
  • Are features (attributes) extracted from the data informative enough for learning?
  • How will success be measured?
  • How will the machine learning solution integrate with other business or research components?

5. Summary

Machine learning is particularly powerful for:

  • Predicting outcomes based on input data, especially when labeled data is available (supervised learning).
  • Discovering patterns or groupings in data where no output labels exist (unsupervised learning).
  • Automating decision-making in contexts ranging from commercial applications like fraud detection, spam classification, and recommendations, to scientific data analysis (e.g., planet detection, DNA sequencing).

The success of machine learning depends on correctly defining the problem, gathering appropriate data, selecting meaningful features, and evaluating models appropriately within the larger context of the problem.

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