AI Mathematicians: How LLMs Are Redefining Mathematics

Author(s): Adit Sheth

Originally published on Towards AI.

What if a computer could discover mathematical patterns, prove theorems, and solve mysteries that have puzzled humanity for centuries? Welcome to the age of AI-powered mathematics, where Large Language Models (LLMs), like ChatGPT, aren’t just assisting mathematicians — they’re rewriting the rules.

Why LLMs Are Changing the Game

Math has always been the domain of human ingenuity. From the Pythagorean theorem to calculus, breakthroughs often required decades of thinking. But here’s the problem: humans are limited by time and cognitive capacity. We can’t test millions of equations or explore trillions of data points overnight.

Enter LLMs:

• Massive Scale: LLMs can process years of mathematical research in seconds.
• Unseen Patterns: They uncover connections in data that humans might miss entirely.
• Creative Proofs: They propose proofs and algorithms in entirely new ways.

Applications of LLMs in Mathematics

Artificial Intelligence is revolutionizing how we approach mathematics. Gone are the days when mathematicians spent years proving a single theorem. Large Language Models (LLMs) like GPT-4 have become groundbreaking tools, uncovering new theorems, solving long-standing problems, and accelerating mathematical research.

Let’s dive into how LLMs are transforming mathematics, with fascinating examples that will leave you amazed.

Prime Gap Analysis

One of the great mysteries of mathematics is understanding the distribution of prime numbers. While primes appear infinite, their spacing becomes less predictable as numbers grow larger. LLMs have contributed by suggesting new conjectures and methods for analyzing prime gaps.

Here’s how LLMs help in prime gap analysis:

• Prime Number Theorem: The prime number theorem gives an approximation of the number of primes less than x.

• Prime Gaps: But what about the gaps between consecutive primes? LLMs, trained on datasets of number theory, can propose formulas and patterns that humans may overlook, guiding mathematicians toward breakthrough insights.

For instance, LLMs can predict gaps between primes for very large numbers. They analyze existing prime number tables and suggest potential new conjectures or relations between gaps that were not previously considered.

Riemann Zeta Function

The Riemann Zeta Function is a cornerstone of number theory. It is defined as:

This function holds the key to one of the greatest unsolved problems in mathematics: the Riemann Hypothesis. LLMs can analyze numerical data and suggest new approaches to test hypotheses about the zeros of the zeta function, something that would take traditional methods years to achieve.

LLMs can:

• Suggest new conjectures on the distribution of non-trivial zeros.
• Generate new ways to numerically approximate these zeros.
• Provide insights into more efficient methods for verifying results.

By analyzing past research and understanding the patterns in zeros of the Riemann zeta function, LLMs can offer breakthroughs in ways humans might not have considered due to the sheer amount of data to process.

Navier-Stokes Equations

The Navier-Stokes equations, used in fluid dynamics, are notoriously difficult to solve. They describe the motion of fluid substances, such as air or water:

While physicists struggle to find analytical solutions for complex fluid behavior, LLMs can simulate and predict flow patterns with unprecedented accuracy.

Here’s how LLMs assist in fluid dynamics:

• Numerical Simulations: LLMs can be trained to simulate real-world fluid behaviors, such as airflow over an airplane wing, water flow in pipes, and more.
• Optimization: They can also optimize fluid-related designs, such as creating more efficient turbines or reducing drag in vehicle designs.
• Time-Saving: LLMs reduce the time needed for manual calculations, making them an invaluable tool for engineers working with complex fluid systems.

For example, LLMs could suggest faster methods for numerically solving these equations in turbulence scenarios that would traditionally take supercomputers hours.

Symbolic Mathematics

Suppose you need the derivative of a polynomial function:

An LLM immediately provides the derivative:

LLMs are also experts at:

• Symbolic Integration: Solving integrals such as

which would be cumbersome for humans to do manually.

• Solving Algebraic Equations: LLMs can handle complex systems of algebraic equations and provide simplified solutions in moments.

LLMs save mathematicians and students significant time by quickly solving these symbolic expressions.

Insights into Heat Transfer

Consider the heat equation, a partial differential equation describing heat distribution over time:

LLMs can solve such equations symbolically or numerically. For instance, in real-world applications like predicting heat flow in metal structures, LLMs help engineers optimize designs much faster than traditional methods.

Here’s how LLMs aid in heat transfer analysis:

• Numerical Simulations: LLMs can simulate heat distribution over time and space for various materials, providing solutions for complex geometries.
• Optimization of Designs: They can propose more efficient materials for heat conduction, saving time in engineering design.
• Real-Time Solutions: They allow engineers to predict heat behavior in real-time for systems that involve dynamic conditions.

By analyzing real-world cases and offering real-time predictions, LLMs significantly improve the speed and accuracy of heat transfer simulations.

Improving Machine Learning with Cross-Entropy Loss

Even machine learning itself benefits from mathematical insights that LLMs provide. The cross-entropy loss function, used in neural networks, is defined as:

Here’s how LLMs improve machine learning:

• Loss Function Variations: LLMs help identify variations of the loss function that can lead to better-performing models.
• Optimization Techniques: By analyzing optimization algorithms, LLMs can suggest more efficient ways to train machine learning models, speeding up convergence.
• Deep Learning Models: In deep learning, LLMs can propose better architectures based on mathematical insights, improving model performance and accuracy.

Revolutionizing Mathematical Research

Here’s how LLMs are reshaping the way math is done:

• Proposing New Theorems: By analyzing historical data, LLMs can predict patterns and propose conjectures that spark new research directions.
• Automating Tedious Calculations: Instead of spending months computing by hand, mathematicians can use LLMs to solve complex integrals, differential equations, and more.
• Generating Insights: LLMs analyze vast mathematical datasets, generating insights that would be impossible to manually uncover, speeding up the discovery of new mathematical properties.

Conclusion: The Future of Mathematical Discovery with LLMs

LLMs are not just tools for automating mundane tasks — they are powerful collaborators in mathematical research. By harnessing their potential, mathematicians can focus on creative problem-solving and exploring new frontiers of knowledge.

• Accelerating Discoveries: LLMs are accelerating the pace of mathematical discoveries, helping us solve problems that once seemed insurmountable.
• Expanding Horizons: They open up new ways to view and solve problems, breaking barriers in various fields of mathematics.

The future of mathematics is no longer just in the hands of human experts; LLMs are transforming how we understand and explore the world of numbers, equations, and beyond.

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Published via Towards AI