Asgard
Write equations. Get differentiable simulations. Reason algebraically.
We have powerful numerical solvers. We have automatic differentiation. We have symbolic algebra systems. What we don't have is a single framework that connects all three — one where you write a mathematical equation, get a differentiable simulation, and can reason about the result algebraically.
Existing tools force a choice. Numerical solvers are fast but opaque. Symbolic systems reason but don't scale. Neural approaches learn but can't be verified. The gap between "define the system" and "understand the system" remains wide, and bridging it requires manual translation between incompatible representations.
Asgard closes this gap. It is the execution and reasoning layer of Gimle — a mathematical formalism where dynamical systems are composed from primitive operators grounded in category theory. You write equations in standard notation. Asgard compiles them into executable, differentiable programs that can be simulated, optimized, and transformed algebraically. The same system runs under deterministic, stochastic, or discrete calculus without changing a line.
How It Works
Write
Define your system as a mathematical equation in LEAN-style notation — a standard, readable syntax used in formal mathematics. Differential equations, integrals, compositions — what you write is what you mean.
Compile
Asgard automatically translates your equation into an executable program built from composable primitive operators. Variable isolation, structural validation, and wiring happen at compile time.
Execute
Run the compiled system under any supported calculus. The same equation produces Taylor-series solutions, Monte Carlo SDE paths, or discrete sequences — depending only on the runtime interpretation.
Reason
Transform, simplify, and prove properties of your system using algebraic rewrite rules. The underlying categorical structure guarantees that every transformation preserves mathematical correctness.
What This Enables
Composable Systems
Build complex dynamics from simple parts. Systems compose sequentially, in parallel, and with feedback — and the composition is algebraically verified, not just bolted together.
Differentiable Everything
Every system compiles to JAX with GPU acceleration. Gradients flow through the entire simulation, enabling parameter optimization, sensitivity analysis, and hybrid physics-ML models out of the box.
One Framework, Many Calculi
Define once, execute under deterministic, stochastic, or discrete semantics. The separation between structure and interpretation means switching calculi is a runtime choice, not a rewrite.
Algebraic Reasoning
A built-in proof system lets you transform and simplify systems mechanically. Find closed-form solutions, verify equivalences, and search for models that fit observed data — all within the same framework.
Hybrid Physics-ML
Compose known physics with learned components. Black-box operators wrap neural networks as first-class building blocks — trainable end-to-end via backpropagation through the full simulation.
Declarative Simulations
Define and run complete simulations from YAML — equation, inputs, evaluation, and visualization — without writing Python. Interactive dashboards render results in real time.
Fields of Application
Climate & Weather
Model coupled atmospheric and ocean dynamics as composable systems. Combine known physics with learned components, and switch between deterministic forecasts and stochastic ensemble simulations.
Finance
Option pricing, risk analysis, and portfolio dynamics as differentiable stochastic systems. Calibrate model parameters via gradient descent and reason about hedging strategies algebraically.
Physics & Engineering
Express governing equations directly and compile to efficient simulations. Compose multi-physics systems, optimize design parameters, and verify structural properties through algebraic transformation.
Control Systems
Feedback loops are first-class citizens. Model controllers, plants, and observers as composable systems with formally verified stability and performance properties.
Biology & Epidemiology
Compartmental models, population dynamics, and gene regulatory networks as composable differential systems. Fit parameters to observed data and explore stochastic mutation pathways.
Scientific Discovery
Search the space of composable dynamical systems for models that explain observed behavior. Algebraic reasoning narrows the search; differentiability enables gradient-guided exploration.
See concrete examples from each of these domains — with the math and the YAML — on the Examples page.