Krish Mehta

Mixture Density Network (MDN)

MDNs output parameters of a probability distribution instead of point predictions. They excel at modeling data with multiple possible outputs for a single input.

Training Data

MDN Predictions

Uncertainty

How it works: The MDN outputs mixture weights (π), means (μ), and standard deviations (σ) for multiple Gaussian components. This allows it to model complex, multi-modal distributions.

p(y|x) = Σᵢ πᵢ(x) · N(y | μᵢ(x), σᵢ²(x))

Physics-Informed Neural Network (PINN)

PINNs incorporate physical laws directly into the neural network training process, enabling them to solve differential equations with limited data.

Boundary Points

PINN Solution

Physics Residual

How it works: PINNs minimize a loss function that includes both data fitting terms and physics constraint terms. The network learns to satisfy the differential equation everywhere in the domain.

Loss = Loss_data + λ · Loss_physics

V-JEPA: Abstract Representation Learning

V-JEPA learns abstract representations by predicting masked video patches in representation space, not pixel space. The model learns to "understand" dynamics rather than "paint" pixels, demonstrating true world understanding through joint embedding predictive architecture.

Context Patches

Masked Patches

Predicted Representations

Target Representations

Key insight: V-JEPA learns in abstract representation space, not pixel space. The Context Encoder processes visible patches, the Predictor predicts target representations, and similarity is measured between predicted and actual representations. This enables understanding of dynamics without pixel-level reconstruction.

Similarity = cosine(Predictor(Context), Target_Encoder(Masked))

30-Year Evolution of Physics Understanding in AI

MDNs - Memorization Era

Probabilistic predictions, massive data needs

PINNs - Rules Era

Physics equations embedded, reduced data

Early World Models

Self-discovered physics from videos

V-JEPA - Understanding Era

Abstract concepts, not pixels

Hybrid Robotics Architecture

Physics Layer

Traditional Simulations

Deterministic Control

Safety Verification

Integration Layer

Decision Engine

Mode Switching

Data Fusion

Intelligence Layer

World Models

Adaptation & Learning

Generalization

When to Use Each Approach

Robotics Task

Precision Critical?

Use Simulation

• Surgery
• Manufacturing
• Safety testing

Novel Environment?

Use World Model

• Home robots
• Exploration
• Human interaction

Complex Real-World Tasks

Use Hybrid System

• Autonomous vehicles
• Warehouse robotics
• Humanoid robots

Training Cost Comparison

Traditional Training

$100k-600k

• Hardware: $50-500k

• Supervision: $50k

• Facility: $10k/mo

• Maintenance: $5-20k

World Model Training

$10k-50k

• Compute: $100k setup

• Data: $10-50k

• Time: 1-2 weeks

• 10x cheaper overall

Industry Adoption Map

Tesla

World models for driving, simulation for safety

Boston Dynamics

Physics for control, AI for terrain adaptation

Google

RT-X models with physics for grasping

2025-2030 Robotics Roadmap

2025-2026

Industry standardization of hybrid pipelines

Edge computing for world models

Regulatory frameworks emerging

→

2027-2028

Digital twins as standard practice

Multi-modal world models

Automated sim-to-real transfer

→

2029-2030

Embodied foundation models

Quantum-enhanced simulation

Self-improving hybrid systems

November 13, 2025

Olivia,

I'm sorry.

I'm sorry that I have failed at giving you the basics. Really, I've been awful at focusing on what matters to me, what truly matters to me these last few months. I've been too wrapped up in other things when I should have been present with you, whenever I'm with you. These last couple days have been hard, and sad for me - I did a lot of reflection, and it made me realize that I haven't really been happy, even though I get to spend time with you.

I want to apologize for not hearing you, for making you feel unheard. I apologize for not showing curiosity — the same curiosity that made you feel special, and made me feel lucky. That said, I also want to redefine "quality time" as I see it, especially because we've had different ideas of what it means. Quality time isn't just time spent together where I learn something new about you, but also something memorable and meaningful — and maybe just us. Time we spent in NYC and Boston was quality time.

I also acknowledge if you feel like I'm not vulnerable enough. It's a flaw of mine. But only with you do I actually feel like being open. You're the first person I've truly opened up to. And still, it's honestly my emotional delay - I will continue working on this.

I hope you haven't completely checked out. I like you so much more than you realize. I'm tired of holding back out of fear, and I hope you feel the same. If you do, can we please talk about it? Miss you

Your stupid boyfriend, Krish