{ "artifacts": [ { "name": "bench_report.json", "url": "https://synthcore.org/code/1215/bench_report.json" }, { "name": "experiment.py", "url": "https://synthcore.org/code/1215/experiment.py" }, { "name": "report.md", "url": "https://synthcore.org/code/1215/report.md" }, { "name": "report_bench_2026-09-03T121411.md", "url": "https://synthcore.org/code/1215/report_bench_2026-09-03T121411.md" }, { "name": "results.json", "url": "https://synthcore.org/code/1215/results.json" }, { "name": "stage2_bench.py", "url": "https://synthcore.org/code/1215/stage2_bench.py" } ], "category": "architecture", "description": "Replace a purely nonlinear recurrent transition with a learned observable map followed by an explicitly linear latent evolution model. Include the original latent state and a small set of nonlinear observables, and update the linear transition online with forgetting-factor recursive least squares when the environment or task dynamics change.", "download_zip": "https://synthcore.org/download/1215", "formulas_latex": [ "$$z_t=\\begin{bmatrix}x_t\\\\\\phi(x_t)\\end{bmatrix},\\qquad z_{t+1}=A_tz_t+B_tu_t+\\epsilon_t,\\qquad \\hat{x}_{t+1}=Cz_{t+1}.$$", "$$\\Theta_t=\\begin{bmatrix}A_t\u0026B_t\\end{bmatrix}=Y_tX_t^{\\top}(X_tX_t^{\\top}+\\alpha I)^{-1},\\quad X_t=\\begin{bmatrix}z_{t-1}\\\\u_{t-1}\\end{bmatrix},\\quad Y_t=z_t.$$", "$$P_t=\\lambda^{-1}\\left[P_{t-1}-\\frac{P_{t-1}r_tr_t^{\\top}P_{t-1}}{\\lambda+r_t^{\\top}P_{t-1}r_t}\\right],\\quad \\Theta_t=\\Theta_{t-1}+\\left(z_t-\\Theta_{t-1}r_t\\right)r_t^{\\top}P_t,$$", "$$\\rho(A_t)\u003c1\\ \\Longrightarrow\\ \\|A_t^k\\|\\leq Kr^k\\ \\text{for some }r\u003c1,\\qquad k_{\\mathrm{adapt}}\\approx\\frac{1}{1-\\lambda}.$$" ], "id": 3063, "implementation": "Use this as the transition module of a compact recurrent state-space model or world model. First encode an observation into x_t=E_psi(o_t), compute z_t=[x_t,phi_eta(x_t)], and predict future states through the linear recurrence z_{t+j+1}=A_t z_{t+j}+B_t u_{t+j}; decode with D_xi. Begin with fixed random Fourier features, polynomial features, or a small two-layer MLP for phi_eta, and train E_psi, D_xi, and phi_eta using one-step and multi-step reconstruction losses. During evaluation, freeze the neural maps and update Theta=[A,B] using the recursive least-squares equations above on every newly observed transition, with P_0=alpha^{-1}I. After each update, estimate the largest eigenvalue of A. If rho(A)\u003er, project eigenvalues outside radius r back to r exp(i arg(mu)), or reject the update and use a damped update. Compute RLS online; estimate residual covariance and spectral radius empirically. The first cheap test is a small CartPole, pendulum, or synthetic nonlinear oscillator dataset with an abrupt mass or force change at the midpoint. Compare a GRU and standard neural SSM against the lifted model for 50-step prediction. The predicted signature is that after the shift, prediction error falls approximately exponentially with an adaptation horizon near 1/(1-lambda); for rho(A)\u003c1, rollout error should decay or saturate, whereas allowing rho(A)\u003e1 should produce an abrupt long-horizon divergence boundary. Sweep transition scaling and verify that measured divergence begins within roughly 20 percent of the predicted rho(A)=1 threshold.", "math_summary": "Let x_t be the encoder state, u_t an optional action or conditioning input, and phi(x_t) a learned or hand-designed observable vector. Define the lifted state z_t=[x_t^T,phi(x_t)^T]^T. The Koopman-inspired model is z_{t+1}=A_t z_t+B_t u_t+epsilon_t, with prediction x_hat_{t+1}=C z_{t+1}, where C selects or decodes the physical state. Estimate A_t and B_t by exponentially weighted least squares with forgetting factor lambda in (0,1]. For discrete-time rollout stability, impose or monitor rho(A_t)\u003c=r\u003c1 after removing known marginal modes; rho is the spectral radius and r is a chosen contraction margin. The adaptation time scale is approximately 1/(1-lambda), while a stable latent component decays geometrically as ||A_t^k|| approximately bounded by K r^k.", "math_tags": [ "dynamical-systems", "linear-algebra", "spectral-theory", "control-theory" ], "ml_areas": [ "ssm", "world-model", "rnn" ], "paper": { "arxiv_id": "2609.02534", "arxiv_url": "https://arxiv.org/abs/2609.02534", "summary_what_math_gives_to_ml": "The paper combines physics-informed observable lifting, finite-dimensional Koopman linearization, online adaptation, and receding-horizon quadratic programming for nonlinear attitude dynamics with abruptly changing inertia and disturbances. Its transferable asset is the mechanism of representing nonlinear dynamics in a lifted coordinate system whose evolution is approximately linear, then updating the lifted dynamics online when the data distribution changes. For neural sequence models and world models, this suggests a latent state-space architecture with explicit linear multi-step propagation, physics-inspired observables, and recursive adaptation of the latent transition operator. The key falsifiable signatures are a measurable spectral stability boundary for long rollouts and an adaptation time scale determined by the forgetting factor.", "title": "Physics-based Online Adaptive Koopman Model Predictive Attitude Control for Combined Spacecraft with Dynamic Uncertainties", "year": "2026" }, "ratings": { "difficulty": 6, "novelty": 6, "usefulness": 8 }, "solves": [ "stability", "accuracy", "generalization" ], "title": "Adaptive Physics-Lifted Koopman State Space", "url": "https://synthcore.org/idea/3063/adaptive-physics-lifted-koopman-state-space", "verification": { "benchmark": { "track": "dynamics", "model": "rnn_small", "n_seeds": 8, "metric_direction": "lower is better", "comparison": { "baseline_mean": 0.0006787698, "idea_mean": 0.000245049, "delta_mean": -0.0004337208, "per_seed_diffs": [ -0.0006494381, -0.0006239988, 0.0003922567, -0.00047326390000000006, -0.00045112250000000004, -0.0008258005, -0.0002562644, -0.0005831346 ], "idea_wins": 7, "n_pairs": 8, "p_value": 0.0231, "mde": 0, "mde_rel_pct": null, "verdict": "idea better (significant)", "system_worked": true }, "baseline": { "best_cfg": { "lr": 0.01 }, "sweep": [ { "cfg": { "lr": 0.001 }, "mean": 0.0032877354 }, { "cfg": { "lr": 0.003 }, "mean": 0.0012035724 }, { "cfg": { "lr": 0.01 }, "mean": 0.0006944253 } ], "full": { "mean": 0.0006787698, "std": 0.0001831161, "per_seed": [ 0.0008624938, 0.0008604837, 0.0003602985, 0.0006422949, 0.0006247925, 0.000911061, 0.0004747614, 0.0006949726 ], "n": 8 } }, "idea": { "mean": 0.000245049, "std": 0.0001979633, "per_seed": [ 0.0002130557, 0.0002364849, 0.0007525552, 0.000169031, 0.00017367, 0.0000852605, 0.000218497, 0.000111838 ], "n": 8 }, "mechanism_signature": { "confirmed": true, "observed_mean_last3_norm_ratio": 0.8155919, "predicted_asymptotic_norm_ratio": 0.7977779, "rho_trained_A": 0.7977779, "source": "trained benchmark Koopman model" }, "custom_track": null }, "mechanism_confirmed": true, "peer_reviewed": false, "practical_verdict": "helps", "stage1_mechanism_check": { "worked": true, "confidence": 8, "verdict": "Built an adaptive lifted Koopman-style transition using [x,x^2] observables and forgetting-factor RLS, with numerical stability and adaptation checks. The RLS recursion exactly matched exponentially weighted batch least squares (max error 1.1e-16), and the measured 63% adaptation time was 14 steps versus the predicted 1/(1-lambda)=14.29. After the dynamics shift, adaptive lifting achieved RMSE 0.0235 in the first 20 steps and 8.28e-5 in the last 100, versus 0.0277 and 0.00379 for adaptive linear; its 50-step rollout MSE was 0.000203 versus 0.00438 frozen. This is a clear toy signal for the claimed adaptation/accuracy effect, though not evidence yet for a general neural world-model win.", "metrics": { "baseline": "Adaptive linear lambda=0.95: post-shift RMSE first20=0.027690, last100=0.003789; frozen lifted 50-step post-shift rollout MSE=0.004381", "idea": "Adaptive lifted [x,x^2] lambda=0.95: post-shift RMSE first20=0.023485, last100=0.0000828; 50-step rollout MSE=0.000203; max measured lifted spectral radius=0.928", "math": "RLS-vs-batch max absolute error=1.11e-16; lambda=0.93 predicted adaptation timescale=14.286, measured 63% timescale=14; scalar rho=0.8 power-29=0.001547" }, "how_to_run": "python3 experiment.py", "files": [ "experiment.py", "results.json" ], "limitations": "Only a deterministic scalar nonlinear synthetic system was tested. The observable map was hand-designed rather than learned, there was no encoder/decoder, GRU or neural SSM baseline, no action-conditioned B matrix evaluation beyond a scalar forcing input, no multi-seed uncertainty estimate, and stability was measured on the fitted lifted matrix without eigenvalue projection or long-duration drifting-regime stress tests." }, "status": "beats_baseline", "status_label": "Beats tuned baseline", "updated_at": "2026-09-03T12:14:11", "verdict_source": "deterministic test code (paired-seed permutation statistics)", "verification_axes": { "benchmark_mechanism": { "confirmed": true, "tested": true }, "practical_benchmark": { "beats_baseline": true, "tested": true, "verdict": "helps" }, "toy_mechanism_gate": { "confirmed": true, "tested": true } } } }