Show HN: Physics written into the architecture (learned Hamiltonian)

A Show HN project called awareliquid_physics released a physics-informed machine learning model that writes a learned Hamiltonian advanced by a symplectic integrator directly into its architecture, conserving energy by construction with bounded O(dt²) drift. In benchmarks, the hard-constraint Hamiltonian head cut 150-step spring energy drift to 0.012 versus 0.036 for an MLP-field Euler control (3.0×) and 100-step orbit energy drift to 6.2 versus 37.8 (6.0×), with orbit rollout MSE of 2.8e-3 versus 6.6e-2 (23.8×). The coupling experiment on a family of oscillators with hidden per-trajectory stiffness showed liquid_ham (liquid core plus Hamiltonian) reaching 4.12 rollout MSE versus 7.24 for a GRU control, while static_ham drifted least at 0.18, which the author reports as a real adaptation/conservation trade-off.

A complete model with physics computation written into the architecture — not bolted on as a loss term. The core idea: a continuous-time liquid LTC recurrence is the substrate, and physical structure a learned Hamiltonian advanced by a symplectic integrator is a hard architectural constraint on top of it. Energy is conserved by construction — bounded O dt² drift, exactly time-reversible — for whatever energy function the network learns. This is the hard-constraint route to physics-informed ML Hamiltonian NNs, Greydanus 2019 , chosen over soft PDE-residual losses after an adversarial review of the PINN literature. observed trajectory prefix │ ▼ LiquidCore LTC multi-timescale gated linear recurrence, │ trained via Blelloch parallel scan O log T ▼ context vector system identification: infers per-trajectory │ hidden parameters e.g. a spring's stiffness ▼ HamiltonianHead separable H q,p = T p + V q | context │ ▼ symplectic rollout velocity-Verlet; energy conserved by construction - awareliquid physics/liquid core.py — the LTC "liquid" recurrence: each timescale is a leaky integrator with a learned time constant; scales are blended by an input-dependent gate. Runs in O log T depth via parallel scan.py . - awareliquid physics/hamiltonian.py — HamiltonianHead hard-constraint, symplectic and MLPFieldHead the honest unstructured control: plain Euler, no conservation . - awareliquid physics/model.py — LiquidHamiltonianModel , the integration the project is about: liquid core reads the prefix → conditions the Hamiltonian's potential → symplectic rollout predicts the future. One fixed learned potential cannot fit a family of systems; the liquid context makes the same model adapt its energy landscape per trajectory. - awareliquid physics/physics ops.py — a zero-parameter, deterministic Newtonian operator layer symplectic Euler integration, N-body gravity, collisions, energy/momentum diagnostics . The ground-truth engine the learned models are benchmarked against: compute, don't memorise . benchmarks/physics rollout eval.py — free-standing head, matched 1-step fit ~5e-6 loss both , scored on physics metrics only: | system | metric | Hamiltonian hard-constraint | MLP-field control | advantage | |---|---|---|---|---| | spring | 150-step energy drift | 0.012 | 0.036 | 3.0× | | orbit | 100-step energy drift | 6.2 | 37.8 | 6.0× | | orbit | 100-step rollout MSE | 2.8e-3 | 6.6e-2 | 23.8× | benchmarks/liquid physics eval.py — the coupling experiment: a family of oscillators with hidden per-trajectory stiffness, prefix → 100-step rollout: | model | rollout MSE | final energy drift | params | |---|---|---|---| | liquid ham liquid + Hamiltonian | 4.12 | 0.70 | 8.7k | | static ham Hamiltonian, no context | 4.41 | 0.18 | 5.0k | | gru seq unstructured control | 7.24 | 2.96 | 10.0k | Honest reading: the physics structure is what buys long-horizon stability over the GRU baseline 2.9× lower rollout error, 4× lower drift ; the liquid context buys adaptation to the hidden parameter better rollout MSE than the static head . The static head drifts least because its one fixed potential is more rigid — the adaptation/conservation trade-off is real and reported as measured. Continuous-state trajectory prediction, validated on physics metrics k-step rollout MSE, energy drift — never perplexity. This is not a language model and makes no hallucination claim. pip install -e ". dev " python -m pytest tests/ -q 13 tests free-standing head: symplectic vs Euler control python benchmarks/physics rollout eval.py --system spring python benchmarks/physics rollout eval.py --system orbit the coupling experiment: liquid ham vs static ham vs gru seq python benchmarks/liquid physics eval.py --train steps 500 All benchmarks are CPU-runnable. - Richer system families 2-body orbits, contacts/collisions via physics ops for the coupling experiment - Non-separable Hamiltonians magnetic / velocity-dependent forces - Scale the liquid substrate; longer-horizon rollouts - Close the adaptation/conservation gap context-conditioned kinetic term, drift-penalised training Note: this repository previously hosted a Kaggle competition entry Nemotron reasoning challenge ; that content lives in the git history before v0.1.0.