This blog is intended for mathematicians with limited background in physics and computational biology.
Recent advancements in data-driven deep learning have transformed mathematics by enhancing—and sometimes surpassing—traditional methods. By leveraging datasets, deep learning techniques are redefining problem-solving and providing powerful tools to tackle challenges once considered impossible. This marks a new paradigm, driven by data, advanced computation, and adaptive learning, pushing the boundaries of what can be achieved. The profound impact of data-driven deep learning was recognized by the 2024 Nobel Prizes in Physics and Chemistry.
The Nobel Prize in Physics honored John Hopfield and Geoffrey Hinton for their groundbreaking contributions to neural networks. Hopfield developed an early model of associative memory in neural networks, known as the Hopfield network, which is based on the concept of energy minimization. The energy function is represented by: \[E(\mathbf{x}) = -\frac{1}{2} \sum_{i, j} w_{ij} x_i x_j + \sum_{i} b_i x_i,\] where $\mathbf{x}$ is the binary state vector, $w_{ij}$ are weights between neurons, and $ b_i$ are biases, This model laid the foundation for optimization in neural networks. The update rule for the state of neuron \( i \) in a Hopfield network can be expressed as: $x_i^{\text{new}} = \text{sign}(\sum_{j} w_{ij} x_j^{\text{old}} + b_i),$ where \( \text{sign}(h) = 1 \) if \( h > 0 \) and \( \text{sign}(h) = -1 \) if \( h \leq 0 \). This rule describes how the network iteratively updates neuron states to minimize the energy function, allowing the network to evolve toward stable states that represent stored memories to optimization problems. The Hopfield network’s reliance on binary states and its tendency to get trapped in local minima both limit its ability to handle more complex tasks.
Hinton developed a stochastic extension of the Hopfield network, retaining the concept of the energy function $E(\mathbf{x})$ but allowing neurons to change states randomly based on probability. Through this stochastic update process, the network gradually moves toward a probability distribution over the states that follows the Boltzmann distribution: \[P(\mathbf{x}) = \frac{e^{-E(\mathbf{x})/T}}{Z},\] where \( T \) is the temperature controlling randomness, and \( Z \) is the partition function ensuring normalization. This extension is known as the
Boltzmann machine. Additionally, Hinton was instrumental in the development of the backpropagation algorithm, which enables neural network training by providing a method to minimize a loss function \( Loss (\mathbf{y}, \mathbf{\hat{y}}) \), where \( \mathbf{y} \) is the true output and \( \mathbf{\hat{y}} \) is the predicted output. The weight update rule in backpropagation is given by: \[w_{ij}^{(new)} = w_{ij}^{(old)} - \eta \frac{\partial Loss}{\partial w_{ij}},\] where \( \eta \) is the learning rate. The backpropagation process involves computing gradients using the chain rule and adjusting the weights to minimize the loss function. These contributions established essential mathematical tools, including activation and loss functions, that continue to shape modern artificial intelligence and problem-solving methods.
The Nobel Prize in Chemistry was awarded to Demis Hassabis, CEO of Google’s DeepMind, and John Jumper, the lead scientist of the AlphaFold project, for their work on the AlphaFold 2 AI model. AlphaFold 2 is an AI-driven tool that predicts the 3D structures of nearly all known proteins based on their amino acid sequences. Previously, computational methods were less accurate, and experimental techniques like X-ray crystallography were time-consuming and costly. AlphaFold dramatically reduces the time and expense associated with determining protein structures and goes beyond traditional homology modeling by accurately predicting structures even for proteins without close structural analogs.
AlphaFold uses graph neural networks and attention mechanisms to model spatial dependencies between amino acids, representing each protein as a graph. In this graph, each node corresponds to an amino acid residue in the sequence, and edges capture interactions and dependencies between amino acids. The objective is to predict the 3D coordinates \( \{\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_n\} \) of residues in a protein sequence of length \( n \). AlphaFold predicts pairwise distances between residues, defined as $d_{ij} = \|\mathbf{r}_i - \mathbf{r}_j\|,$ where \( \mathbf{r}_i \) and \( \mathbf{r}_j \) are the 3D coordinates of residues \( i \) and \( j \). These distances are modeled with distogram distributions $P(d_{ij}) \approx \text{Distogram}(d_{ij})$ to capture uncertainties. The model uses multi-head attention mechanisms (inspired by the transformer architecture) to capture both local and long-range dependencies across residues. For each residue \( i \), the attention mechanism computes interactions with other residues \( j \) in the sequence, defined as:
\[\text{Attention}(\mathbf{h}_i) = \sum_{j } \alpha_{ij} \mathbf{W}_v \mathbf{h}_j,\]
where \( \alpha_{ij} = \frac{\exp(\mathbf{q}_i^\top \mathbf{k}_j)}{\sum_{k \in V} \exp(\mathbf{q}_i^\top \mathbf{k}_k)} \) is the attention weight, \( \mathbf{q}_i = \mathbf{W}_q \mathbf{h}_i \) and \( \mathbf{k}_j = \mathbf{W}_k \mathbf{h}_j \) are query and key projections, \( \mathbf{W}_q \), \( \mathbf{W}_k \), and \( \mathbf{W}_v \) are learned weight matrices.The model minimizes an energy-like loss function, which penalizes deviations from known structural constraints, expressed as:
\[ Loss = \sum_{i < j} \|d_{ij}^{\text{pred}} - d_{ij}^{\text{true}}\|^2 + \lambda \sum_{k} \| \theta_k^{\text{pred}} - \theta_k^{\text{true}} \|^2, \]
where \( d_{ij}^{\text{pred}} \) and \( d_{ij}^{\text{true}} \) are the predicted and true pairwise distances between residues, \( \theta_k^{\text{pred}} \) and \( \theta_k^{\text{true}} \) represent predicted and true bond angles, and \( \lambda \) is a weighting factor. This loss function is minimized using backpropagation and gradient descent, iteratively refining the model to achieve high structural accuracy.
Other AI breakthroughs have addressed previously insurmountable challenges, including Transformer models in natural language processing, reinforcement learning in AlphaGo and AlphaZero, supervised learning for medical image enhancement, GANs for realistic data generation, and AI in climate modeling and weather prediction.
While AI has proven effective in standardized, data-rich environments, certain limitations emerge when applied to healthcare. Medical data often presents diverse and nuanced patterns that current AI systems may struggle to interpret accurately. For instance, while ChatGPT has advanced language processing, IBM Watson Health faced challenges in converting clinical data into personalized cancer treatments. This underscores AI’s reliance on statistical patterns, which can limit its ability to handle clinical uncertainty or recognize subtle signs that experienced doctors instinctively detect. Although I respect Hinton’s significant contributions to AI, I disagree with his overestimations about AI’s future, particularly the notion that it will replace doctors.
Generative models, though promising in imaging, reveal specific limitations in medical applications. Trained to reconstruct images from learned patterns, these models may misinterpret critical but uncommon anomalies as noise or generate “hallucinations” in out-of-distribution cases, potentially leading to overlooked abnormalities or misdiagnosis. Their tendency to favor common patterns can also cause them to miss rare features that are essential for diagnosis.
These limitations suggest that AI functions best as a complementary tool in healthcare. It can enhance diagnostic capabilities and assist clinicians in specific tasks, but it cannot fully replicate the depth of expertise and interpretive skill that healthcare professionals bring. By combining AI with traditional methods, we can achieve outcomes that neither approach could reach independently, positioning AI as a valuable asset in advancing healthcare.
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