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| id: ML25 |
| title: "Reservoir computing vs MLP vs GP for short-horizon Lorenz-63 forecasting" |
| arxiv_id: null |
| venue: "ARC-Bench 2026" |
| paper_asset: null |
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| synthesis: | |
| Reservoir computing (a.k.a. echo state networks, ESN) is a lightweight |
| approach for learning dynamical systems: a fixed random recurrent network |
| projects the input into a high-dimensional reservoir state, and only a |
| linear readout layer is trained. For chaotic systems, ESNs have repeatedly |
| been reported to match or outperform trainable recurrent nets on |
| short-horizon trajectory prediction, despite training only a ridge |
| regression on the readout. |
| |
| A credible CPU-scale study of this claim must compare (i) a reservoir |
| network with a random sparse internal matrix and fixed spectral radius, |
| (ii) a parameter-matched fully-connected MLP that reads a fixed window of |
| past states, and (iii) a Gaussian Process regressor with an RBF kernel on |
| the same windowed input. The target signal is Lorenz-63 integrated at |
| dt=0.02 with the classical (σ=10, ρ=28, β=8/3) parameters. Training-set |
| sizes N ∈ {200, 500, 1000, 2000} are small enough that the standard |
| reservoir-computing folklore — "ESNs win when data is scarce" — can |
| actually be tested rather than assumed. |
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| The key dynamics-aware metric is the *valid prediction time* (VPT): |
| the first time step at which the normalized prediction error exceeds a |
| threshold (commonly 0.4 of the attractor standard deviation). Unlike |
| per-step RMSE, VPT tracks how long a model's forecast remains useful on |
| the chaotic attractor. Reporting both RMSE and VPT reveals whether any |
| observed "superiority" is numerical or dynamical. |
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| The research question is: *does a small reservoir (N_res ≤ 500 units) beat |
| a matched-parameter MLP and a Gaussian Process on valid prediction time |
| for Lorenz-63, and at which training-set size does the gap appear?* |
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| hypotheses: |
| - id: H1 |
| statement: "An echo-state network with spectral radius ≈ 0.9 and ≤ 500 reservoir units achieves a longer mean Valid Prediction Time (VPT ≥ 1.2×) than a parameter-matched MLP on Lorenz-63 at training-set sizes N ≤ 500." |
| measurable: true |
| - id: H2 |
| statement: "The Gaussian Process regressor is competitive with (VPT within ±20% of) the ESN at N=200 but is decisively beaten (ESN VPT at least 2× the GP's) at N=2000, reflecting the GP's O(N^3) training bottleneck and the ESN's ability to exploit more data without re-inverting a Gram matrix." |
| measurable: true |
| - id: H3 |
| statement: "Per-step RMSE at one-step-ahead prediction is NOT a reliable proxy for VPT: at least one condition exists where model A has lower one-step RMSE than model B but a shorter VPT, showing that per-step error understates the divergence on chaotic attractors." |
| measurable: true |
|
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| experiment_design: |
| research_question: "On Lorenz-63 short-horizon forecasting, does a small echo-state network achieve a longer valid prediction time than a parameter-matched MLP and a Gaussian Process regressor, and how does the ranking depend on training-set size?" |
| conditions: |
| - name: "esn_N200" |
| description: "Echo-state network with 300 reservoir units, spectral radius 0.9, input scaling 1.0, leak rate 0.3, ridge regression readout (alpha=1e-5). Trained on N=200 consecutive Lorenz-63 samples." |
| - name: "esn_N500" |
| description: "Same ESN configuration, N=500 training samples." |
| - name: "esn_N1000" |
| description: "Same ESN configuration, N=1000 training samples." |
| - name: "esn_N2000" |
| description: "Same ESN configuration, N=2000 training samples." |
| - name: "mlp_N{200,500,1000,2000}" |
| description: "Fully-connected MLP with 2 hidden layers whose total parameter count is within ±10% of the ESN readout parameter count. Reads a 5-step window of past (x,y,z) states as input. Adam optimizer, early stopping on a 10% validation split." |
| - name: "gp_N{200,500,1000,2000}" |
| description: "Gaussian Process regressor with an RBF kernel and WhiteNoise term, hyperparameters optimised by marginal likelihood maximisation. Reads the same 5-step window as the MLP. At N=2000 the GP may time out or run out of memory; record the failure." |
| baselines: |
| - "Persistence baseline: predicted next state = previous state. Establishes a floor for VPT." |
| - "Linear autoregressive baseline: one-step-ahead linear regression on the 5-step window. Establishes whether the non-linear methods' gains are real." |
| metrics: |
| - name: "valid_prediction_time" |
| direction: "maximize" |
| description: "First forecast step (in units of integration dt) at which the normalized L2 error exceeds 0.4 × attractor std. Averaged over 10 random initial conditions on an independent test trajectory." |
| - name: "one_step_rmse" |
| direction: "minimize" |
| description: "Root-mean-square error of one-step-ahead prediction on the test trajectory." |
| - name: "train_wall_clock_sec" |
| direction: "minimize" |
| description: "Wall-clock time to fit the model, including hyperparameter optimisation where applicable." |
| - name: "param_count" |
| direction: "report" |
| description: "Number of trainable parameters. Must be reported to demonstrate that the MLP is parameter-matched to the ESN." |
| datasets: |
| - name: "lorenz63_train" |
| source: "synthetic — integrate dxdt=σ(y-x), dy/dt=x(ρ-z)-y, dz/dt=xy-βz at dt=0.02 with σ=10, ρ=28, β=8/3. Use scipy.integrate.solve_ivp or a hand-rolled RK4." |
| - name: "lorenz63_test" |
| source: "independent Lorenz-63 trajectory (different IC, 10× longer warmup) with 10 evaluation windows for VPT averaging." |
| compute_requirements: |
| gpu_required: false |
| estimated_wall_clock_sec: 720 |
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| rubric_path: "experiments/arc_bench/config/ml/rubrics/ML25.json" |
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