papers/luno/sau.json

{
  "paper_id": "luno",
  "paper_title": "LUNO: Linearized Uncertainty Quantification for Neural Operators",
  "D1": [
    {
      "id": "luno-D1-001",
      "claim": "FNO architecture uses 4 Fourier blocks with 12 Fourier modes per spatial dimension, 18 hidden dimensions, 10 input time steps, autoregressive single-step output, and 2-point zero-padding on the spatial grid.",
      "source": "Section 2.1, Example 2.1, Section 5"
    },
    {
      "id": "luno-D1-002",
      "claim": "Low-data training regime: 25 training trajectories, 100 epochs, AdamW optimizer with cosine decay LR schedule and warmup, MSE loss, 250 validation and 250 test input-output pairs.",
      "source": "Section 5, Appendix D, Table 3"
    },
    {
      "id": "luno-D1-003",
      "claim": "OOD training regime: 1000 training trajectories (Base), 1000 epochs, AdamW optimizer with cosine decay LR schedule and warmup, MSE loss, 250 validation and 250 test input-output pairs per dataset.",
      "source": "Section 5, Appendix D"
    },
    {
      "id": "luno-D1-004",
      "claim": "Low-data PDE datasets: Burgers, Hyper Diffusion, Kuramoto-Sivashinsky (conservative), all 1D with spatial resolution 256, temporal resolution 59, 25 train / 250 valid / 250 test trajectories.",
      "source": "Section 5, Appendix D, Table 3"
    },
    {
      "id": "luno-D1-005",
      "claim": "OOD advection-diffusion-reaction datasets: diffusion coefficient alpha=0.026, 100x100 spatial grid, dt=5e-10, 200 total time steps subsampled to 59, 1-10 random Gaussian blobs as initial conditions, custom 9-point stencil with RK4 solver, 5 datasets (Base, Flip, Pos, Pos-Neg, Pos-Neg-Flip), 1000 Base train trajectories with 250 valid and 250 test pairs each.",
      "source": "Section 5, Appendix D, Figure 5"
    },
    {
      "id": "luno-D1-006",
      "claim": "Deep ensemble uses 10 independently trained FNO members with different random seeds for entirely separate training runs.",
      "source": "Section 5"
    },
    {
      "id": "luno-D1-007",
      "claim": "LUNO uncertainty methods: low-rank GGN approximation rank 500; GGN computed on all input-output pairs (25 trajectories x 59 steps) for low-data or minibatch of 1000 pairs for OOD; 200 sample-based push-forward samples; calibration via grid search over 500 logarithmically spaced points minimizing expected marginal NLL; calibrated hyperparameter is sigma^2 (variance); weight-space prior is isotropic Gaussian N(0, sigma_prior^2 I); linearization point is MAP estimate w*.",
      "source": "Section 3.2, Section 3.2.1, Section 5, Appendix B"
    },
    {
      "id": "luno-D1-008",
      "claim": "Compared UQ methods: Input Perturbations, Deep Ensemble, Sample-Iso, LUNO-Iso, Sample-LA, LUNO-LA; last-layer Laplace scope limited to last Fourier block parameters (R^{(L-1)}, W^{(L-1)}) only.",
      "source": "Section 5, Section 3.2.1"
    },
    {
      "id": "luno-D1-009",
      "claim": "Evaluation metrics: RMSE, chi-squared calibration statistic (ideal=1.0, >1 overconfidence, <1 underconfidence), marginal NLL; 250 test pairs; 1.96 std multiplier for confidence intervals.",
      "source": "Section 5, Appendix D"
    },
    {
      "id": "luno-D1-010",
      "claim": "Autoregressive rollout evaluation uses 50 trajectories from each OOD dataset (Base, Flip, Pos, Pos-Neg, Pos-Neg-Flip), 10 initial time steps, iterative single-step prediction over 59 total time steps with sliding window of 10.",
      "source": "Section 5, Figure 4, Appendix D, Figure 7"
    }
  ],
  "D2": [
    {
      "id": "luno-D2-001",
      "claim": "FNO Lifting (Input Projection): v^{(1)}(x) = p(a(x), w_p) in R^{d_v'}, maps input function from dimension d_A' to hidden dimension d_v' via parametric function p (linear layer or shallow MLP).",
      "source": "Section 2.1, Example 2.1"
    },
    {
      "id": "luno-D2-002",
      "claim": "FNO Fourier Layer Update: v_i^{(l+1)}(x) = sigma^{(l)}( sum_j F^{-1}( (R_{kij}^{(l)} F(v_j^{(l)})_k )_{k=1}^{k_max} )(x) + W_{ij}^{(l)} v_j^{(l)}(x) ), combining spectral convolution (Fourier domain multiplication + inverse FFT) with local linear transform and element-wise nonlinearity sigma^{(l)}. Iterated for layers l = 1,...,L-1.",
      "source": "Section 2.1, Example 2.1"
    },
    {
      "id": "luno-D2-003",
      "claim": "FNO Projection (Output Mapping): F(a,w)(x) = q(v^{(L)}(x), w_q), projects final hidden representation back to output function dimension d_U' via parametric function q.",
      "source": "Section 2.1, Example 2.1"
    },
    {
      "id": "luno-D2-004",
      "claim": "Empirical Risk (Training Loss): R(w) = (1/n) sum_i L(u^{(i)}(X_U^{(i)}), F(a^{(i)}(X_A^{(i)}), w)(X_U^{(i)})), standard MSE loss summed over spatial grid points, minimized via AdamW with cosine decay LR.",
      "source": "Section 2.1, Eq 2.1"
    },
    {
      "id": "luno-D2-005",
      "claim": "Negative Log-Posterior (Bayesian Training Objective): R(w) = -log p(w) - sum_i log p(y^{(i)} | f(x^{(i)}, w)) + const, with Gaussian prior N(0, sigma_prior^2 I) giving L2 regularization. MAP estimate w* = argmin_w R(w) is the trained network weights.",
      "source": "Section 3.2, Step 2"
    },
    {
      "id": "luno-D2-006",
      "claim": "Model Linearization (First-Order Taylor): f_mu^{lin}((a,x), w) = f((a,x), mu) + D_w f((a,x), w)|_mu (w - mu), linearizes neural operator around mean weights mu (typically w*) to enable analytic uncertainty propagation.",
      "source": "Section 3.2, Step 2, Eq 3"
    },
    {
      "id": "luno-D2-007",
      "claim": "Linearized GP Mean Function: m(a,x) = f((a,x), mu) = F(a, mu)(x), the mean prediction equals the original neural operator's output at MAP weights.",
      "source": "Section 3.2, Step 2"
    },
    {
      "id": "luno-D2-008",
      "claim": "Linearized GP Covariance Function: K((a1,x1), (a2,x2)) = D_w f((a1,x1), w)|_mu * Sigma * D_w f((a2,x2), w)|_mu^T, the Jacobian-sandwiched weight-space covariance. This is the fundamental formula for linearized uncertainty propagation, inducing a d_U'-output GP.",
      "source": "Section 3.2, Step 3, Eq 4"
    },
    {
      "id": "luno-D2-009",
      "claim": "Uncurrying Map: f: (A x D_U) x W -> R^{d_U'}, ((a,x), w) |-> F(a,w)(x). Converts the operator F (mapping input functions to output functions) into a standard neural network f (mapping augmented inputs (a,x) to vector outputs), enabling application of standard Bayesian deep learning tools.",
      "source": "Section 3.2, Step 1"
    },
    {
      "id": "luno-D2-010",
      "claim": "Probabilistic Currying (Function-Valued GP Reconstruction): F: A x Omega -> U, (a, omega) |-> (x |-> f((a,x), omega)). Reinterprets the multi-output GP f as a function-valued GP F where F(a) is itself a GP over the output domain D_U.",
      "source": "Section 3.1, Theorem 3.2, Section 3.2, Step 3"
    },
    {
      "id": "luno-D2-011",
      "claim": "Function-Valued GP Predictive Mean and Covariance: E[F(a)(x)] = F(a, mu)(x); Cov[F(a1)(x1), F(a2)(x2)] = D_w F(a1,w)(x1)|_mu * Sigma * D_w F(a2,w)(x2)|_mu^T. The expected value equals the trained operator's prediction; covariance expressed in terms of the neural operator's Jacobian.",
      "source": "Section 3.2, Step 3, Eq 4"
    },
    {
      "id": "luno-D2-012",
      "claim": "Generalized Gauss-Newton (GGN) Matrix: G = -sum_i D_w f(x^{(i)}, w)|_{w*} * H_f log p(y^{(i)} | f)|_{f(x^{(i)}, w*)} * D_w f(x^{(i)}, w)|_{w*}^T. Approximates the Hessian of negative log-likelihood. For Gaussian likelihood, H_f log p = I/sigma_obs^2. Low-rank (rank 500) via Lanczos.",
      "source": "Section 3.2, Appendix B"
    },
    {
      "id": "luno-D2-013",
      "claim": "Laplace Posterior Precision and Approximation: P = -H_w log p(w)|_{w*} + G. Laplace posterior p(w|D) ~= N(w; w*, P^dagger), with Moore-Penrose pseudoinverse used when P is degenerate (low-rank GGN). Total precision = prior precision + GGN.",
      "source": "Section 3.2, Step 2"
    },
    {
      "id": "luno-D2-014",
      "claim": "LLA Posterior Predictive GP: f|D ~ GP( f(., w*), (x1,x2) |-> D_w f(x1,w)|_{w*} * P^dagger * D_w f(x2,w)|_{w*}^T ). Under the linearized model, the Laplace posterior over weights induces a tractable GP posterior predictive.",
      "source": "Section 3.2, Step 2"
    },
    {
      "id": "luno-D2-015",
      "claim": "FNO Last-Layer Factorization: F(a,w)(x) = q_tilde( z^{(L-1)}(x, w_{L-1}) ), with q_tilde = q(., w_q) o sigma^{(L-1)} and w_{L-1} = (R^{(L-1)}, W^{(L-1)}). Factorizes FNO into a linear (in w_{L-1}) feature map z^{(L-1)} followed by a nonlinear projection q_tilde, enabling efficient last-layer Laplace.",
      "source": "Section 3.2.1, Appendix C.1, Eq 5"
    },
    {
      "id": "luno-D2-016",
      "claim": "z^{(L-1)} Fourier Feature Decomposition: z_i^{(L-1)}(x, w_{L-1}) = sum_{j,k} Re(R_{kij}^{(L-1)}) * Re(v_hat_{kj}^{(L-1)}) * cos(<omega_k, x>) + sum_{j,k} (-1) * Im(R_{kij}^{(L-1)}) * Im(v_hat_{kj}^{(L-1)}) * sin(<omega_k, x>) + sum_j W_{ij}^{(L-1)} * v_j^{(L-1)}(x). Explicit real-valued expansion showing z^{(L-1)} is linear in weights w_{L-1}.",
      "source": "Section 3.2.1, Appendix C.1"
    },
    {
      "id": "luno-D2-017",
      "claim": "Weight Reparameterization for Last-Layer GP: w_{L-1} ~= (Re(R^{(L-1)}), Im(R^{(L-1)}), W^{(L-1)}) flattened into a real-valued parameter vector, enabling standard Gaussian belief modeling over all last-layer parameters.",
      "source": "Section 3.2.1, Appendix C.1"
    },
    {
      "id": "luno-D2-018",
      "claim": "z^{(L-1)} GP Induced by Weight Uncertainty: z^{(L-1)} ~ GP(m_{z^{(L-1)}}, K_{z^{(L-1)}}), with m_{z^{(L-1)}}(x) = z^{(L-1)}(x, w_{L-1}^*). Because z^{(L-1)} is linear in w_{L-1}, Gaussian weight uncertainty induces a multi-output GP over z with parametric Fourier basis features.",
      "source": "Section 3.2.1, Appendix C.1"
    },
    {
      "id": "luno-D2-019",
      "claim": "Last-Layer LUNO Function-Valued GP: F(a)(x) = q_tilde(m_{z^{(L-1)}}(x)) + D q_tilde(m_{z^{(L-1)}}(x)) * (z^{(L-1)}(x) - m_{z^{(L-1)}}(x)). Applies a second linearization (of q_tilde around m_z) to push the z-GP through the nonlinear projection, yielding the final function-valued GP.",
      "source": "Section 3.2.1, Appendix C.1, Eq 5"
    },
    {
      "id": "luno-D2-020",
      "claim": "Last-Layer LUNO Predictive Mean and Covariance: m_a(x) = F(a, w*)(x); K_a(x1, x2) = D q_tilde(m_{z^{(L-1)}}(x1)) * K_{z^{(L-1)}}(x1, x2) * D q_tilde(m_{z^{(L-1)}}(x2))^T. Mean equals deterministic FNO output. Covariance is z-GP covariance sandwiched by q_tilde Jacobians — efficient since only small network Jacobian required.",
      "source": "Section 3.2.1, Appendix C.1"
    },
    {
      "id": "luno-D2-021",
      "claim": "Isotropic Gaussian Weight Prior: w ~ N(w*, Sigma = sigma^2 I), the simplest weight-space uncertainty model with scalar variance sigma^2 as the sole calibration hyperparameter, used in LUNO-Iso and Sample-Iso methods.",
      "source": "Section 3.2, Section 5"
    },
    {
      "id": "luno-D2-022",
      "claim": "Low-Rank GGN Push-Forward Distribution: F(a,w)(X) ~ N( F(a,w*)(X), J_theta * Sigma * J_theta^T ), with Sigma = (n V V^T + sigma I)^{-1} where V V^T is the rank-500 GGN approximation. Retains only the 500 largest eigenmodes with diagonal prior term for full-rank stability.",
      "source": "Section 3.2, Section 5, Appendix B"
    },
    {
      "id": "luno-D2-023",
      "claim": "Evaluation Metrics: RMSE = sqrt((1/n) sum_i (y_i - y_hat_i)^2); Marginal NLL = -sum_i log(1/sqrt(2*pi*sigma_i^2) * exp(-(y_i - y_hat_i)^2 / (2*sigma_i^2))); Chi-Squared Q = (1/n) sum_i (y_i - y_hat_i)^2 / sigma_i^2, with Q~1 indicating well-calibrated uncertainty, Q>1 overconfidence, Q<1 underconfidence.",
      "source": "Section 5, Appendix D"
    },
    {
      "id": "luno-D2-024",
      "claim": "Input Perturbation Gaussian Noise: epsilon_{x,t} ~ N(0, sigma^2), additive perturbation applied to each input grid point. An ensemble of perturbed forward passes estimates predictive moments empirically. Baseline method calibrated on validation NLL.",
      "source": "Section 5"
    }
  ],
  "D3": [
    {
      "id": "luno-D3-001",
      "claim": "Low-Data UQ Evaluation Protocol: Evaluate predictive uncertainty quantification methods (LUNO vs baselines) on FNOs trained in a low-data regime with only 25 trajectories from each of 3 PDEs (Burgers, Hyper Diffusion, Kuramoto-Sivashinsky), testing whether linearized uncertainty (LUNO-LA) outperforms sample-based, ensemble, and input perturbation approaches. Metrics: RMSE, chi^2, marginal NLL on 250 test pairs. Calibration: grid search over 500 logarithmically spaced sigma^2 values minimizing validation NLL.",
      "source": "Section 5, Table 1, Figure 2, Appendix D, Table 4, Table 5"
    },
    {
      "id": "luno-D3-002",
      "claim": "OOD Robustness Evaluation Protocol: Assess robustness of UQ methods under distribution shift by training FNO on 1000 in-distribution advection-diffusion-reaction Base trajectories and testing on 4 OOD datasets (Flip: velocity reversed; Pos: triangular heat source; Pos-Neg: heat source + cloud-shaped sink; Pos-Neg-Flip: source + sink + velocity flip). Custom PDE solver with 9-point stencil and RK4, 100x100 grid, dt=5e-10, diffusion coefficient alpha=0.026. Primary metric: marginal NLL. Training: 1000 epochs on Base only, GGN computed on minibatch of 1000 pairs, all hyperparameters calibrated on 250 Base validation pairs.",
      "source": "Section 5, Table 2, Figure 3, Appendix D, Tables 6-11"
    },
    {
      "id": "luno-D3-003",
      "claim": "Autoregressive Rollout Evaluation Protocol: Evaluate UQ methods under full-trajectory autoregressive roll-out where predictions are recursively fed back as inputs (sliding window of 10 steps, 59 total time steps), causing accumulated errors and distribution shift over time. 50 rollout trajectories from each OOD dataset. Metrics: RMSE and marginal NLL per time step. Compares how uncertainty estimates adapt to accumulating errors.",
      "source": "Section 5, Figure 4, Appendix D, Figure 7"
    },
    {
      "id": "luno-D3-004",
      "claim": "Hyperparameter Calibration Protocol: Calibrate the variance hyperparameter sigma^2 for all UQ methods via grid search on validation NLL. For each method, search over 500 logarithmically spaced sigma^2 values, compute predictive distribution on 250 validation pairs, compute expected marginal NLL, select sigma^2 that minimizes NLL, fix for final test evaluation.",
      "source": "Section 5"
    },
    {
      "id": "luno-D3-005",
      "claim": "Runtime Comparison Protocol: Compare computational efficiency of linearization-based (LUNO) vs sampling-based (Sample) UQ methods for iterative single-trajectory autoregressive rollout. Measure wall-clock time per trajectory rollout with mean and standard deviation. Implementation in JAX leveraging Jacobian-vector products and analytical IFFT Jacobian structure for LUNO methods.",
      "source": "Section 5, Appendix D, Table 12"
    },
    {
      "id": "luno-D3-006",
      "claim": "FNO Training Protocol: Train Fourier Neural Operators on PDE trajectory data for next-step prediction. Architecture: 4 Fourier blocks, 12 modes, 18 hidden dimensions. Input: 10 consecutive time steps + velocity field + reaction term glued and zero-padded by 2 grid points. Optimizer: AdamW with cosine decay LR schedule and warmup. Loss: MSE. Low-data: 100 epochs (one pass per trajectory pair); OOD: 1000 epochs. Framework: JAX + Flax NNX + optax. Trained weights w* saved as MAP estimate.",
      "source": "Section 2.1, Example 2.1, Section 5"
    }
  ],
  "D4": [
    {
      "id": "luno-D4-001",
      "claim": "Low-Data UQ Evaluation Step Sequence: (1) Generate PDE datasets via APEBench (25 train + 250 valid + 250 test trajectories per PDE); (2) Train FNO for 100 epochs using AdamW+cosine decay on 25 trajectories; (3) Compute MAP estimate w* as trained weights; (4) For Laplace methods: compute low-rank GGN approximation (rank 500) using all training pairs; (5) For LUNO methods: linearize FNO around w*, push weight-space Gaussian to function-space GP; (6) For Sample methods: draw 200 weight samples, push through nonlinear FNO, estimate moments; (7) For Ensemble: train 10 FNOs independently, aggregate predictions; (8) Calibrate sigma^2 via grid search on validation NLL; (9) Evaluate RMSE, chi^2, NLL on 250 test pairs; (10) Visualize predictive mean, standard deviation, samples, eigenfunctions, and covariance.",
      "source": "Section 5, Table 1, Figure 2, Appendix D"
    },
    {
      "id": "luno-D4-002",
      "claim": "OOD Robustness Evaluation Step Sequence: (1) Generate 5 advection-diffusion-reaction datasets with custom PDE solver (RK4 + 9-point stencil); (2) Train FNO for 1000 epochs on 1000 Base trajectories; (3) Compute Laplace posterior: low-rank GGN (rank 500) on minibatch of 1000 pairs; (4) Calibrate all UQ method hyperparameters on 250 Base validation pairs via NLL grid search; (5) Evaluate all methods on 250 test pairs from each OOD dataset (Base, Flip, Pos, Pos-Neg, Pos-Neg-Flip); (6) Compute RMSE, chi^2, NLL per method per dataset; (7) Compare ensemble null-space residual projection vs LUNO full-rank covariance.",
      "source": "Section 5, Table 2, Figure 3, Appendix D"
    },
    {
      "id": "luno-D4-003",
      "claim": "Autoregressive Rollout Evaluation Step Sequence: (1) Take trained FNO and calibrated UQ methods from low-data or OOD experiments; (2) For each method, run autoregressive rollout on 50 test trajectories; (3) At each step, feed model prediction back as input (sliding window of 10); (4) Compute RMSE and NLL per time step; (5) Plot performance degradation over rollout horizon; (6) Compare how uncertainty estimates adapt to accumulating errors.",
      "source": "Section 5, Figure 4, Figure 7, Appendix D"
    },
    {
      "id": "luno-D4-004",
      "claim": "Hyperparameter Calibration Step Sequence: (1) For each UQ method, define a logarithmically spaced grid of 500 sigma^2 values; (2) For each candidate sigma^2, compute predictive distribution on 250 validation pairs; (3) Compute expected marginal NLL; (4) Select sigma^2 that minimizes NLL; (5) Fix calibrated sigma^2 for final test evaluation.",
      "source": "Section 5"
    },
    {
      "id": "luno-D4-005",
      "claim": "Runtime Comparison Step Sequence: (1) Load trained FNO and calibrated UQ methods; (2) For each method, run full autoregressive rollout on one trajectory; (3) Measure wall-clock time; (4) Repeat for statistical robustness; (5) Report mean +/- standard deviation.",
      "source": "Section 5, Table 12, Appendix D"
    },
    {
      "id": "luno-D4-006",
      "claim": "FNO Training Step Sequence: (1) Initialize FNO with 4 Fourier blocks (12 modes, 18 hidden dims); (2) Prepare training batches: 10 input time steps, predict next step; (3) Glue velocity field and reaction term to input; (4) Pad input by 2 constant zero grid points to reduce boundary artifacts; (5) Train with AdamW + cosine decay LR schedule with warmup; (6) Minimize MSE loss over specified epochs; (7) Save trained weights w* as MAP estimate.",
      "source": "Section 2.1, Example 2.1, Section 5"
    }
  ]
}