papers/wdno/sau.json

{
  "paper_id": "wdno",
  "paper_title": "Wavelet Diffusion Neural Operator (WDNO)",
  "D1": [
    {
      "id": "wdno-D1-001",
      "claim": "WDNO BRM/SRM training config (all experiments): Adam optimizer, lr=1e-4, cosine annealing scheduler, 190000 training steps, batch_size=16. 1D experiments on 1 A100 GPU (2.4-2.5h), 2D on 2 A100 GPUs (7.8-7.9h).",
      "source": "Table 18, Table 19, Table 20, Table 12"
    },
    {
      "id": "wdno-D1-002",
      "claim": "WDNO 1D U-Net architecture hyperparameters (for Burgers/Advection/Navier-Stokes): init_dim=128, down_up_layers=4, conv_kernel=3, dim_multiplier=[1,2,4,8], resnet_block_groups=8, attn_hidden_dim=32, attn_heads=4.",
      "source": "Table 18, Table 19"
    },
    {
      "id": "wdno-D1-003",
      "claim": "1D DDIM sampling config: DDIM_sampling_steps=50, DDIM_eta=1 (equivalent to DDPM stochastic sampling) for Burgers, Advection, and Navier-Stokes inference.",
      "source": "Table 18, Table 19"
    },
    {
      "id": "wdno-D1-004",
      "claim": "1D Burgers control guidance config: guidance_weight=120000 with cosine scheduler for guidance intensity during control optimization.",
      "source": "Table 18, Section C.4 Table 8"
    },
    {
      "id": "wdno-D1-005",
      "claim": "1D Burgers PDE and dataset config: T=8, D=[0,1], nu=0.01 Dirichlet BC; 40000 train trajectories; test: 50 (control), 2000 (super-res 0x), 100 (super-res 1x/2x/3x); 81 stored timesteps; state shape [81,120], force shape [80,120]; solver internal grid 120x16 spatial, 4800x16 temporal, downsample factor 16.",
      "source": "Appendix F.1, Appendix F.2"
    },
    {
      "id": "wdno-D1-006",
      "claim": "1D Burgers initial condition generation: sum of 2 Gaussians; amplitudes a1~U(0,2), a2~U(-2,0); positions b1~U(0.2,0.4), b2~U(0.6,0.8); widths sigma~U(0.05,0.15).",
      "source": "Appendix F.2"
    },
    {
      "id": "wdno-D1-007",
      "claim": "1D Burgers control force generation: sum of 8 Gaussians; spatial/temporal positions b1,i~U(0,1), b2,i~U(0,1); widths sigma1,i~U(0.1,0.4), sigma2,i~U(0.1,0.4); amplitude a1~U(-1.5,1.5); for i>=2, ai~U(-1.5,1.5) or 0 with 50% probability.",
      "source": "Appendix F.2"
    },
    {
      "id": "wdno-D1-008",
      "claim": "1D wavelet transform config (Burgers/Advection/Navier-Stokes): 2D DWT using bior2.4 wavelet, periodization padding mode, pytorch_wavelets (Cotter 2019). Output shape [4,41,60] from 81x120 input. Applies to 81 timesteps (Burgers) and 80 timesteps (Advection, from PDEBench).",
      "source": "Appendix F.3, Appendix G.2"
    },
    {
      "id": "wdno-D1-009",
      "claim": "1D Navier-Stokes dataset config: shear and bulk viscosity eta=zeta=1e-8, heat capacity ratio Gamma=5/3, data resolution [81,120], 9000 training samples, PDEBench CFD shock dataset ('1D_CFD_Shock_Eta1.e-8_Zeta1.e-8_trans_Train.hdf5').",
      "source": "Section 4.3, Appendix G.1"
    },
    {
      "id": "wdno-D1-010",
      "claim": "2D wavelet transform config (Fluid/ERA5): 3D DWT using bior1.3 wavelet, zero padding mode, implemented via ptwt (Wolter 2024). Input shape [32,64,64], output wavelet coefficient shape [8,18,34,34] (1 coarse + 7 detail channels).",
      "source": "Section 4.4, Appendix H.2"
    },
    {
      "id": "wdno-D1-011",
      "claim": "2D incompressible fluid dataset config: 32 time steps, 64x64 spatial grid, 3584 peripheral control variables per time step, fluid-solid coupling with no-slip boundary conditions at obstacles.",
      "source": "Section 4.4"
    },
    {
      "id": "wdno-D1-012",
      "claim": "WDNO 3D U-Net hyperparameters (2D experiments): Conv3D kernel=(3,3,3), padding=(1,1,1), stride=(1,1,1); spatial-only downsampling kernel=(1,4,4), stride=(0,1,1); spatial-only upsampling kernel=(1,4,4); attn_heads=4. Inspired by Video Diffusion Models (Ho et al., 2022).",
      "source": "Table 20"
    },
    {
      "id": "wdno-D1-013",
      "claim": "2D DDIM sampling config: DDIM_sampling_steps_2D=100, DDIM_eta_2D=1, used for 2D fluid and ERA5 inference.",
      "source": "Table 20"
    },
    {
      "id": "wdno-D1-014",
      "claim": "2D fluid control guidance config: guidance_weight_2D=100 for indirect smoke navigation control task.",
      "source": "Table 20"
    },
    {
      "id": "wdno-D1-015",
      "claim": "Zero-shot super-resolution config: 1D base [80,120], targets [160,240] (1x), [320,480] (2x), [640,960] (3x); 2D base [32,64,64], target [32,128,128]. Multi-res wavelet coeff shapes: 1D [4x21x30, 4x11x15, 4x6x8]; 2D [8x18x18x18, 8x18x10x10].",
      "source": "Section 4.6, Appendix F.3, Appendix H.2"
    },
    {
      "id": "wdno-D1-016",
      "claim": "ERA5 dataset config: 0.25-degree latitude-longitude resolution, surface to 100km altitude, variable=temperature. Prediction task: 12h input history to forecast 20h ahead.",
      "source": "Section 4.5"
    },
    {
      "id": "wdno-D1-017",
      "claim": "Ablation study configs: measurement noise scales [0.01, 0.001, 0.0001] times data_std on 1D Burgers; training sample fractions [0.2, 0.4, 0.6, 0.8] of 9000 samples on 1D Navier-Stokes; control sequence noise prob=0.1 in robustness test.",
      "source": "Section 4.7, Table 9, Section C.5"
    },
    {
      "id": "wdno-D1-018",
      "claim": "SAC baseline hyperparameters (1D Burgers control): discount=0.5, target_smoothing=0.05, critic_lr=3e-4, entropy_lr=3e-3, policy_lr=3e-3, batch=8192, episodes=1500, model_updates_per_step=50, target_updates=15, replay_buffer=1M, energy_weight=2e-5.",
      "source": "Table 22"
    },
    {
      "id": "wdno-D1-019",
      "claim": "BPPO baseline hyperparameters (1D Burgers control): state value net (lr=1e-4, 2M steps, 3 layers, hidden=512); Q net (lr=1e-4, 2M steps, 2 layers, hidden=1024, discount=0.99, target_update=2, soft_update=0.005); BC pretrain (lr=1e-4, 500k episodes); main (100k episodes, 2 layers, hidden=1024, lr=1e-5, clip=0.25, ReLU, batch=512).",
      "source": "Table 24"
    },
    {
      "id": "wdno-D1-020",
      "claim": "BC baseline hyperparameters (1D Burgers control): lr=1e-4, batch=512, episodes=500k, replay_buffer=2M, 2 layers, hidden=1024, ReLU activation.",
      "source": "Table 25"
    },
    {
      "id": "wdno-D1-021",
      "claim": "SL (Supervised Learning) baseline hyperparameters (1D Burgers control): LBFGS optimizer, lr=0.1, epochs=100, objective loss weight=1, reconstruction loss weight=0.01, tolerance=1e-5.",
      "source": "Table 23"
    },
    {
      "id": "wdno-D1-022",
      "claim": "ANN PID baseline hyperparameters (1D Burgers control): 1D conv kernel=3, padding=1, stride=1, Softsign activation, batch=16, lr=1e-4, MAE loss.",
      "source": "Table 21"
    },
    {
      "id": "wdno-D1-023",
      "claim": "FNO baseline hyperparameters (1D and 2D): 1D: modes=16, width=64, in_ch=3, out_ch=1, 4 Fourier layers, lift/proj hidden=256, GeLU activation, lr=1e-4; 2D: modes=16, width=64, in_ch=6, out_ch=4, 4 Fourier layers, lr=1e-4, epochs=1000, cosine scheduler.",
      "source": "Table 26, Table 32"
    },
    {
      "id": "wdno-D1-024",
      "claim": "WNO baseline hyperparameters (1D and 2D): 1D: sym4 wavelet, 5 levels, uplift_dim=40, 4 layers, batch=100, lr=1e-3, epochs=1000, StepLR scheduler; 2D: db4 wavelet, 2 levels, uplift_dim=8, 3 layers, batch=50, lr=0.05, epochs=500, StepLR scheduler.",
      "source": "Table 27, Table 33"
    },
    {
      "id": "wdno-D1-025",
      "claim": "MWT baseline hyperparameters (1D and 2D): 1D: Legendre wavelet, 10 Fourier modes, kernel=4, batch=256, epochs=300, MultiStepLR scheduler; 2D: Legendre wavelet, 12 Fourier modes, kernel=3, batch=200, epochs=300, MultiStepLR scheduler.",
      "source": "Table 29, Table 34"
    },
    {
      "id": "wdno-D1-026",
      "claim": "OFormer 1D baseline: SpatialEncoder2D (in_ch=3, embed_dim=96, encoded_dim=256, heads=4, depth=6, res=120, dropout=0.05), PointWiseDecoder2DSimple (latent=256, out_ch=1), batch=32, 50000 iterations, lr=1e-4.",
      "source": "Table 30"
    },
    {
      "id": "wdno-D1-027",
      "claim": "OFormer 2D baseline: SpatialTemporalEncoder2D (in_ch=3, embed_dim=496, encoded_dim=192, heads=1, depth=5), PointWiseDecoder2D (embed_dim=96, out_ch=1, out_seq_len=32, propagate_forward=1, curriculum_ratio=0.1, curriculum_steps=6), batch=8, 100000 iterations, lr=1e-4.",
      "source": "Table 35"
    },
    {
      "id": "wdno-D1-028",
      "claim": "MS-L-NODE (MSVI) baseline hyperparameters (1D simulation): encoder CNN channels=128, latent_dim=8, aggregation heads=1, 6 static + 4 dynamic layers, batch=64, 37500 iterations, lr=1e-3.",
      "source": "Table 31"
    },
    {
      "id": "wdno-D1-029",
      "claim": "CNN 1D baseline: conv kernel=5, padding=2, ELU activation, latent=256, batch=100, lr=1e-3, epochs=500, cosine scheduler.",
      "source": "Table 28"
    },
    {
      "id": "wdno-D1-030",
      "claim": "Sensitivity analysis configs: DDIM steps grid [20,40,50,100,200]; DDIM eta grid [0.2,0.5,0.8,1.0] for 1D Burgers control; guidance weight grid [9,10,11.5,12.5,13] x1e4 for 2D control.",
      "source": "Table 7, Table 8"
    }
  ],
  "D2": [
    {
      "id": "wdno-D2-001",
      "claim": "DDPM forward process (noise addition): x_{k+1} = sqrt(alpha_k) * x_k + sqrt(1 - alpha_k) * epsilon, where epsilon ~ N(0, I) and {alpha_k}_{k=1..K} is the variance schedule. Implemented via: x_k = sqrt(alpha_bar_k) * x_0 + sqrt(1 - alpha_bar_k) * epsilon, where alpha_bar_k = prod_{i=1..k} alpha_i.",
      "source": "Section 2.2"
    },
    {
      "id": "wdno-D2-002",
      "claim": "DDPM reverse process (denoising): x_{k-1} = 1/sqrt(alpha_k) * (x_k - (1-alpha_k)/sqrt(1-alpha_bar_k) * epsilon_theta(x_k, k)) + sigma_k * z, where z ~ N(0, I) if k > 1 else z = 0. epsilon_theta is the learned denoising U-Net.",
      "source": "Section 2.2"
    },
    {
      "id": "wdno-D2-003",
      "claim": "DDPM training loss (simplified variational bound): L = E_{k~U(1,K), x_0~p(x), epsilon~N(0,I)} [|| epsilon - epsilon_theta(sqrt(alpha_bar_k) * x_0 + sqrt(1 - alpha_bar_k) * epsilon, k) ||_2^2]",
      "source": "Section 2.2, Eq. 3"
    },
    {
      "id": "wdno-D2-004",
      "claim": "Classifier-free guidance sampling: epsilon_theta_guided(x, y) = epsilon_theta(x, empty) + omega * (epsilon_theta(x, y) - epsilon_theta(x, empty)), where omega in [0, 1] is the guidance weight and 'empty' is the null-condition identifier.",
      "source": "Section 2.2"
    },
    {
      "id": "wdno-D2-005",
      "claim": "Discrete wavelet transform (1D signal decomposition): phi_{l,m}(x) = 2^{l/2} * phi(2^l * x - m); psi_{l,m}(x) = 2^{l/2} * psi(2^l * x - m). With l_0 = L (maximum level), the decomposition is: u(x) = sum_m c_L(m) * phi_{L,m}(x) + sum_m d_L(m) * psi_{L,m}(x), where c_L are coarse coefficients and d_L are detail coefficients. In code: apply 2D DWT via pytorch_wavelets (1D) or 3D DWT via ptwt (2D) to get coefficient arrays.",
      "source": "Section 3.1"
    },
    {
      "id": "wdno-D2-006",
      "claim": "WDNO simulation: denoising in wavelet domain: Initialize W_{u,[0,T]}^{(K)} ~ N(0, I). For k = K down to 1: W_{u,[0,T]}^{(k-1)} = W_{u,[0,T]}^{(k)} - eta * epsilon_theta(W_{u,[0,T]}^{(k)}, W_a, k) + xi, where xi ~ N(0, sigma_k^2 * I) if k > 1 else xi = 0. W_a is the wavelet-transformed equation parameter (e.g., initial condition). Final output is obtained by inverse DWT on W_{u,[0,T]}^{(0)}. DDIM acceleration is used during inference.",
      "source": "Section 3.1, Eq. 3"
    },
    {
      "id": "wdno-D2-007",
      "claim": "WDNO control: guided denoising with objective gradient: Initialize W_{f,[0,T]}^{(K)} ~ N(0, I). For k = K down to 1: W_{f,[0,T]}^{(k-1)} = W_{f,[0,T]}^{(k)} - eta * (epsilon_theta(W_{f,[0,T]}^{(k)}, W_a, k) + lambda * grad_{W_f} J(W_hat_{f,[0,T]}^{(k)})) + xi, where xi ~ N(0, sigma_k^2 * I) if k > 1 else xi = 0. W_hat_{f,[0,T]}^{(k)} is the estimated noise-free W_f^{(0)} extracted via: W_hat_{f,[0,T]}^{(k)} = (W_{f,[0,T]}^{(k)} - sqrt(1 - alpha_bar_k) * epsilon_theta(W_{f,[0,T]}^{(k)}, W_a, k)) / sqrt(alpha_bar_k). lambda is the guidance weight. J is the control objective.",
      "source": "Section 3.1, Eq. 4, Eq. 5"
    },
    {
      "id": "wdno-D2-008",
      "claim": "WDNO noise-free estimate from noisy sample (one-step prediction): W_hat_f^{(k)} = (W_f^{(k)} - sqrt(1 - alpha_bar_k) * epsilon_theta(W_f^{(k)}, W_a, k)) / sqrt(alpha_bar_k). This is used to compute the objective J on the estimated clean data rather than noisy data, avoiding noise-induced errors in the guidance gradient.",
      "source": "Section 3.1, Eq. 5"
    },
    {
      "id": "wdno-D2-009",
      "claim": "Multi-resolution training: dataset preparation by downsampling: Given original resolution N x M (time x space), create data pairs by downsampling: (N, M) and (N/2, M/2); (N/2, M/2) and (N/4, M/4); (N/4, M/4) and (N/8, M/8); etc. Apply wavelet transform to each resolution level. For 1D Burgers with original [80, 120]: downsampled to [40, 60], [20, 30], [10, 15]. Wavelet coefficients: original 81x120 -> 4x41x60; downsampled 41x60 -> 4x21x30; 21x31 -> 4x11x15; 11x15 -> 4x6x8.",
      "source": "Section 3.2, Appendix F.3"
    },
    {
      "id": "wdno-D2-010",
      "claim": "Super-Resolution Model (SRM) training: Train conditional diffusion model p(W_h | W_l, W_{a_h}) using paired high-resolution W_h and low-resolution W_l data. To align sizes, duplicate low-resolution coefficients to match high-resolution dimensions (with boundary duplication for odd numbers). Each training batch randomly selects data pairs from a given resolution level. Training loss is the standard DDPM MSE loss between predicted and true noise.",
      "source": "Section 3.2"
    },
    {
      "id": "wdno-D2-011",
      "claim": "WDNO zero-shot super-resolution inference: 1) Downsample a to base resolution NxM, wavelet transform to get W_a. 2) Use BRM to generate wavelet coefficients W_base at resolution NxM. 3) Use SRM iteratively: condition on W_current (low-res) and W_{a_next} (high-res a at next level) to generate W_next at 2Nx2M. 4) Repeat SRM until target resolution reached. 5) Apply inverse wavelet transform to get final trajectory.",
      "source": "Section 3.2"
    },
    {
      "id": "wdno-D2-012",
      "claim": "1D Burgers control objective function J: J = integral_D |u(T,x) - u*(x)|^2 dx + alpha * integral_{[0,T]xD} |f(t,x)|^2 dt dx. The first term penalizes deviation from target state at final time. The second term penalizes control energy. In code: MSE between u_pred(T) and u_target, plus alpha * MSE of control force f over all time steps.",
      "source": "Section 4.1, Eq. 6, Appendix F.1"
    },
    {
      "id": "wdno-D2-013",
      "claim": "1D Burgers initial condition generation: u(0,x) = sum_{i=1}^{2} a_i * exp(-(x - b_i)^2 / (2 * sigma_i^2)). Parameters sampled: a_1 ~ U(0,2), a_2 ~ U(-2,0), b_1 ~ U(0.2, 0.4), b_2 ~ U(0.6, 0.8), sigma_i ~ U(0.05, 0.15).",
      "source": "Appendix F.2"
    },
    {
      "id": "wdno-D2-014",
      "claim": "1D Burgers control force generation: f(t,x) = sum_{i=1}^{8} a_i * exp(-(x - b_{1,i})^2 / (2 * sigma_{1,i}^2)) * exp(-(t - b_{2,i})^2 / (2 * sigma_{2,i}^2)). Parameters: b_{1,i} ~ U(0,1), b_{2,i} ~ U(0,1), sigma_{1,i} ~ U(0.1, 0.4), sigma_{2,i} ~ U(0.1, 0.4), a_1 ~ U(-1.5, 1.5), and for i >= 2: a_i ~ U(-1.5, 1.5) or 0 with equal probability.",
      "source": "Appendix F.2"
    },
    {
      "id": "wdno-D2-015",
      "claim": "1D Burgers finite difference solver: Finite difference method solving: du/dt = -u * du/dx + nu * d^2u/dx^2 + f(t,x), with Dirichlet boundary u=0, initial u(0,x)=u_0(x), nu=0.01. Internal grid: 120x16 spatial, 4800x16 temporal steps. Control f is kept constant between two stored time stamps. After simulation, downsample by factor 16 spatially and temporally before saving.",
      "source": "Appendix F.1, F.2"
    },
    {
      "id": "wdno-D2-016",
      "claim": "2D wavelet data preparation for WDNO (1D Burgers/Advection/Navier-Stokes): 1) Apply 2D DWT to trajectory data (81x120) using bior2.4 wavelet, periodization mode -> outputs 4 coeff sets each (41x60): 1 coarse + 3 detail. 2) Apply 1D DWT to initial condition (1D), repeat coeffs, concatenate with 2D coeffs. 3) For multi-resolution data: downsample data to 41x60, 21x31, 11x15; apply 2D DWT -> 4x21x30, 4x11x15, 4x6x8. 4) Align low-res coeffs to high-res by duplication, with boundary duplication for odd dimensions.",
      "source": "Appendix F.3"
    },
    {
      "id": "wdno-D2-017",
      "claim": "3D wavelet data preparation for WDNO (2D incompressible fluid): 1) Apply 3D DWT to trajectory data (32x64x64) using bior1.3 wavelet, 'zero' padding mode -> outputs 8 coeff sets each (18x34x34): 1 coarse + 7 detail. 2) Apply 2D DWT to initial condition (2D) and 1D DWT to smoke percentage, repeat coeffs and concatenate. 3) For multi-resolution: downsample to 32x32x32, 32x16x16; apply 3D DWT -> 8x18x18x18, 8x18x10x10.",
      "source": "Appendix H.2"
    },
    {
      "id": "wdno-D2-018",
      "claim": "WDNO training pseudocode: Training: 1) Apply DWT to u_{[0,T]}^{(0)} ~ q(u) to get W_u^{(0)}. 2) Sample k ~ Uniform(1,...,K). 3) Sample epsilon ~ N(0,I). 4) Take gradient descent step on ||epsilon - epsilon_theta(sqrt(alpha_bar_k) * W_u^{(0)} + sqrt(1-alpha_bar_k) * epsilon, k)||^2. Repeat until converged. Sampling (simulation): 1) W_u^{(K)} ~ N(0,I). 2) For k=K..1: W_u^{(k-1)} = W_u^{(k)} - eta * epsilon_theta(W_u^{(k)}, W_a, k) + xi (xi~N(0,I) if k>1 else 0). 3) Apply inverse DWT to W_u^{(0)} to get u_{[0,T]}^{(0)}. For control: same but with additional +lambda * grad J(W_hat_f^{(k)}) term in step 2.",
      "source": "Algorithm 1 (Appendix E)"
    },
    {
      "id": "wdno-D2-019",
      "claim": "1D Navier-Stokes sound speed and Mach number computation: Sound speed: c_s = sqrt(Gamma * p / rho), where Gamma=5/3. Mach number: M = |v| / c_s. Initial velocity field: v(x, t=0) = sum_{i=1..n} A_i * sin(k_i * x + phi_i), where A_i = v_bar / |k|^d, d=1 for 2D case, v_bar = c_s * M. Shock-tube initialization: Q(x, t=0) = (Q_L, Q_R) with piecewise constant values generating shocks and rarefactions, where Q = (rho, v, p).",
      "source": "Appendix G.1"
    },
    {
      "id": "wdno-D2-020",
      "claim": "SAC reward function for 1D control: r(t, u_t, u_d, w_t) = -integral_Omega |u_t - u_d|^2 dx - alpha * integral_Omega |w_t|^2 dx. In code: negative MSE between current state and target state minus alpha * MSE of action. Alpha is energy weight (0.00002 for Burgers).",
      "source": "Appendix I.2, Eq. 13"
    },
    {
      "id": "wdno-D2-021",
      "claim": "BPPO reward function for 1D control: r(t, u_t, u_d, f_t) = -integral_Omega |u_t - u_d|^2 dx - alpha * integral_Omega |f_t|^2 dx. Same form as SAC reward with state u and control f.",
      "source": "Appendix I.4, Eq. 14"
    },
    {
      "id": "wdno-D2-022",
      "claim": "Approximate scale invariance for multi-resolution alignment: Given high-res data d_+ at NxM and low-res data d_- at (N/2)x(M/2), apply linear transformation t_scaled = a_1 * t_orig + b_1, x_scaled = a_2 * x_orig + b_2 to rescale coordinates. Then the stretched function satisfies: du/(a_1*dt) = F(u, du/(a_2*dx), d^2u/(a_2^2*dx^2), ...) + f(t,x). Since resolution change factor is constant, a_1 and a_2 are fixed, so the pattern between resolutions is consistent. This consistency also holds in wavelet domain due to linearity and locality of wavelet transform.",
      "source": "Section 3.2, Eq. 7"
    },
    {
      "id": "wdno-D2-023",
      "claim": "WDNO U-Net architecture (2D encoder-decoder for 1D spatiotemporal data): Two separate U-Nets: epsilon_theta(f) for force pathway and epsilon_theta(u,f) for state pathway. Each U-Net: initial dim=128, 4 down/up layers with 3x3 conv kernels, channel multipliers [1,2,4,8] per stage, 8 ResNet block groups per stage, attention at lowest resolution with hidden dim=32 and 4 heads.",
      "source": "Table 18, Appendix F.4"
    },
    {
      "id": "wdno-D2-024",
      "claim": "WDNO 3D U-Net architecture (for 2D spatiotemporal data): Three modules: downsampling encoder, middle module, upsampling decoder with 3D spatiotemporal convolutions. Conv3D: kernel=(3,3,3), padding=(1,1,1), stride=(1,1,1). Downsampling: kernel=(1,4,4), padding=(1,2,2), stride=(0,1,1) - only downsamples spatial dims. Upsampling: kernel=(1,4,4), padding=(1,2,2), stride=(0,1,1) - only upsamples spatial dims. 4 attention heads. Architecture inspired by Video Diffusion Models (Ho et al., 2022).",
      "source": "Table 20, Appendix H.3"
    },
    {
      "id": "wdno-D2-025",
      "claim": "Guidance loss for initial condition and target state consistency: Add auxiliary loss terms to the control guidance gradient: ∇_{W_f} J_total = ∇_{W_f} J_control + λ_IC · ∇_{W_f} ||IDWT(W_f[0,:]) - u_0_true||^2 + λ_target · ∇_{W_f} ||IDWT(W_f[T,:]) - u_T_true||^2, where W_f[0,:] is the initial timestep's wavelet coefficient channel, W_f[T,:] is the target timestep's wavelet coefficient channel, IDWT(·) is the inverse discrete wavelet transform, and u_0_true, u_T_true are the ground truth initial/target states. This ensures the generated trajectory matches the given initial condition and target state. For 1D Burgers control: λ_IC applied via cosine-scheduled guidance weight (120000); for 2D fluid control: λ_target = 100.",
      "source": "Appendix F.4, Appendix H.3"
    },
    {
      "id": "wdno-D2-026",
      "claim": "DDIM accelerated sampling: Denoising Diffusion Implicit Model (DDIM) (Song et al., 2020) is used during inference to speed up sampling. Instead of running all K diffusion steps, DDIM uses a subsequence of steps. WDNO uses 50 DDIM iterations for 1D and 100 for 2D inference, with eta=1 (equivalent to DDPM stochastic sampling).",
      "source": "Section 3.1, Table 18, Table 20"
    },
    {
      "id": "wdno-D2-027",
      "claim": "ANN PID controller architecture (1D baseline): Two 1D convolutional layers (kernel=3, padding=1, stride=1) with Softsign activation, followed by two fully connected layers and four activation layers. Trained with MAE loss between current and target states using Adam optimizer (lr=0.0001, batch=16). The neural network outputs MIMO PID parameters based on error Err_t, which the PID controller uses to produce control f_t.",
      "source": "Appendix I.1, Table 21"
    },
    {
      "id": "wdno-D2-028",
      "claim": "Supervised Learning control algorithm (SL baseline): Stage 1: Two CNN VAEs project state u and control f into latent space; a CNN learns latent transition u_t -> u_{t+1}. Stage 2: Use these three networks as surrogate models to compute gradient of objective J w.r.t. control f via backpropagation. Optimize f with LBFGS (lr=0.1, 100 epochs, tol=1e-5). Include reconstruction loss for control (weight=0.01) alongside objective loss (weight=1.0) to avoid adversarial modes.",
      "source": "Appendix I.3, Table 23"
    },
    {
      "id": "wdno-D2-029",
      "claim": "FNO architecture (1D baseline): Fourier Neural Operator: Lifting block (hidden=256) -> 4 Fourier layers (modes=16, width=64) -> Projection block (hidden=256). Each Fourier layer: FFT -> linear transform in Fourier domain on kept modes -> IFFT -> add skip connection. MLP expansion=0.5, GeLU activation, rank of tensor factorization=1.0, one-sided domain padding. Input: initial state + controls (3 channels). Output: states (1 channel). Supports zero-shot super-resolution.",
      "source": "Appendix J.1, Table 26"
    },
    {
      "id": "wdno-D2-030",
      "claim": "WNO architecture (1D baseline): Wavelet Neural Operator: sym4 wavelet, 5 decomposition levels, uplifting dimension=40, 4 wavelet layers. Each wavelet layer: DWT -> linear transform on wavelet coefficients -> IDWT + skip connection. Trained with Adam (lr=1e-3, batch=100, epochs=1000, StepLR scheduler).",
      "source": "Appendix J.2, Table 27"
    },
    {
      "id": "wdno-D2-031",
      "claim": "MWT architecture (1D baseline): Multiwavelet Neural Operator: Legendre wavelet basis, 10 Fourier modes, kernel size=4. Batch=256, epochs=300, Adam optimizer, MultiStepLR scheduler.",
      "source": "Appendix J.4, Table 29"
    },
    {
      "id": "wdno-D2-032",
      "claim": "OFormer architecture (1D baseline): Operator Transformer: SpatialEncoder2D (3 input channels, embedding dim=96, encoded sequence dim=256, 4 heads, 6 depth layers, resolution=120, dropout=0.05) -> PointWiseDecoder2DSimple (latent channels=256, output channels=1, scale=0.5, resolution=120). Batch=32, 50000 iterations, lr=1e-4.",
      "source": "Appendix J.5, Table 30"
    }
  ],
  "D3": [
    {
      "id": "wdno-D3-001",
      "claim": "1D Burgers equation simulation: Learn mapping from initial condition u_0 and force f to entire trajectory u_{[0,T]}. Compare WDNO against neural operator baselines (WNO, MWT, OFormer, FNO, CNN, DDPM). Metrics: MSE on state sequences excluding initial conditions. Datasets: 40000 trajectories with random initial conditions (2 Gaussians) and control forces (8 Gaussians).",
      "source": "Section 4.1, Table 1, Appendix F"
    },
    {
      "id": "wdno-D3-002",
      "claim": "1D Burgers equation control: Find optimal control force f that minimizes objective J = integral|u(T,x)-u*(x)|^2 + alpha*integral|f|^2. Compare WDNO against PID, SAC (pseudo-online/offline), BPPO (surrogate-solver/solver), BC (surrogate-solver/solver), SL, and DDPM baselines. Metrics: control objective J. Datasets: 40000 train / 50 test trajectories.",
      "source": "Section 4.1, Table 2a, Appendix F"
    },
    {
      "id": "wdno-D3-003",
      "claim": "1D Advection equation simulation: Predict 80 timesteps of advection evolution from a one-step initial condition. Evaluate on smooth dynamics from PDEBench. Compare against WNO, MWT, OFormer, FNO, CNN, DDPM. Data prep: 2D wavelet transform (bior2.4, periodization). Metric: MSE excluding initial conditions.",
      "source": "Section 4.2, Table 1"
    },
    {
      "id": "wdno-D3-004",
      "claim": "1D Compressible Navier-Stokes equation simulation: Evaluate on shock-tube scenarios from PDEBench with extremely small viscosity (eta=1e-8, zeta=1e-8). Simulation only (no control). Compare against Transolver, CNO, MSVI, ACDM, DiffusionPDE, WNO, MWT, OFormer, FNO, CNN, DDPM, Diffusion+FFT, FNO Denoiser. Metrics: MSE, MAE, L_inf. Datasets: 9000 training samples, 81x120 resolution.",
      "source": "Section 4.3, Table 1, Table 5, Appendix G"
    },
    {
      "id": "wdno-D3-005",
      "claim": "2D Incompressible fluid simulation: Predict smoke density, velocity field, and percentage through target bucket given initial smoke density and control sequences. Fluid-solid coupling with no-slip boundaries at obstacles. Compare against WNO, MWT, OFormer, FNO, U-Net, DDPM. Data prep: 3D DWT (bior1.3, zero mode). Metric: MSE excluding initial conditions. Datasets: 32 timesteps, 64x64 spatial grid.",
      "source": "Section 4.4, Table 1, Appendix H"
    },
    {
      "id": "wdno-D3-006",
      "claim": "2D Incompressible fluid control (indirect smoke navigation): Control 3584 peripheral variables over 32 timesteps to steer smoke from below central obstacle into top-center bucket. Objective J = percentage of smoke NOT passing target bucket. Compare against BC, BPPO, SAC (pseudo-online/offline), DDPM. Highly challenging due to indirect control and need for trajectory planning.",
      "source": "Section 4.4, Table 2b, Appendix H"
    },
    {
      "id": "wdno-D3-007",
      "claim": "ERA5 weather simulation: Predict temperature evolution over next 20 hours given past 12 hours of atmospheric data. Real-world dataset (0.25deg, surface to 100km). Compare against MWT, OFormer, FNO, U-Net, DDPM. Data prep: 3D DWT (bior1.3). Metrics: MSE, Relative L2 error.",
      "source": "Section 4.5, Table 1"
    },
    {
      "id": "wdno-D3-008",
      "claim": "Zero-shot super-resolution (1D Burgers): Train on 80x120; evaluate at 160x240 (1x), 320x480 (2x), 640x960 (3x) unseen during training. Compare WNO, FNO, WDNO with linear and nearest interpolation. Test: 2000 trajectories (0x) + 100 shared (1x/2x/3x). Metric: MSE at finest resolution.",
      "source": "Section 4.6, Table 16, Figure 4"
    },
    {
      "id": "wdno-D3-009",
      "claim": "Zero-shot super-resolution (2D fluid): Train on 32x64x64; evaluate at 32x128x128 unseen during training. Compare FNO and WDNO with linear and nearest interpolation. Metric: MSE at finest resolution.",
      "source": "Section 4.6, Table 17, Figure 4"
    },
    {
      "id": "wdno-D3-010",
      "claim": "Ablation: wavelet transform for abrupt changes. Compare WDNO vs DDPM MAE per time step specifically at regions with abrupt spatial changes in 1D Burgers and 1D Navier-Stokes. Demonstrate wavelet benefit at shock/discontinuity regions.",
      "source": "Section 4.7, Figure 6, Figure 9"
    },
    {
      "id": "wdno-D3-011",
      "claim": "Ablation: wavelet + multi-resolution training synergy. Compare WDNO (wavelet domain) vs DDPM with multi-resolution training in original space-time domain (no wavelet). Evaluate at multiple super-resolution levels. Metric: MSE at finest resolution.",
      "source": "Section 4.7, Figure 4c"
    },
    {
      "id": "wdno-D3-012",
      "claim": "Ablation: wavelet vs Fourier transform. Compare WDNO (wavelet domain generation) vs Diffusion+FFT (Fourier domain, same architecture/pipeline except transform). Also test FNO as denoiser backbone. Dataset: 1D Navier-Stokes. Metric: MSE.",
      "source": "Section 4.7, Figure 5c"
    },
    {
      "id": "wdno-D3-013",
      "claim": "Ablation: long-term dependency. Measure error growth over time steps for WDNO vs WNO, MWT, OFormer, FNO, U-Net, DDPM on 2D simulation. Verify slower error accumulation for WDNO. Metric: MSE per time step.",
      "source": "Section 4.7, Figure 5a"
    },
    {
      "id": "wdno-D3-014",
      "claim": "Ablation: measurement noise robustness. Add Gaussian noise (scale factors 0.01, 0.001, 0.0001 * data_std) to both training and testing data of 1D Burgers. Compare WDNO vs DDPM. Metric: MSE vs noise scale.",
      "source": "Section 4.7, Figure 5d"
    },
    {
      "id": "wdno-D3-015",
      "claim": "Ablation: training sample size. Reduce training set to 0.2, 0.4, 0.6, 0.8 of full 9000 samples on 1D Navier-Stokes. Measure WDNO MSE trend. Metric: MSE.",
      "source": "Section 4.7, Figure 5b"
    },
    {
      "id": "wdno-D3-016",
      "claim": "Ablation: approximate scale invariance verification. Train FNO on original, once-downsampled, twice-downsampled, and mixed datasets; test at all three resolutions to demonstrate mixed training improves generalization. Metric: MSE at each resolution.",
      "source": "Appendix C.3, Table 6"
    },
    {
      "id": "wdno-D3-017",
      "claim": "Sensitivity analysis: DDIM sampling steps. Test steps in {20, 40, 50, 100, 200} for 1D Burgers simulation and 2D fluid control. Metrics: MSE (1D sim), J (2D control).",
      "source": "Appendix C.4, Table 7, Table 8"
    },
    {
      "id": "wdno-D3-018",
      "claim": "Sensitivity analysis: DDIM eta and guidance weight. Test eta in {0.2, 0.5, 0.8, 1.0} for 1D Burgers control; test guidance weights {9, 10, 11.5, 12.5, 13} x1e4 for 2D control. Metrics: MSE, J.",
      "source": "Appendix C.4, Table 7, Table 8"
    },
    {
      "id": "wdno-D3-019",
      "claim": "Robustness test: control sequence noise. 1D Burgers control with 0.1 probability of noise in control during evaluation. Compare WDNO vs PID, SAC, BC, BPPO, SL, DDPM. Metric: J.",
      "source": "Appendix C.5, Table 9"
    },
    {
      "id": "wdno-D3-020",
      "claim": "Robustness test: variance across test samples. Report mean +/- std of control objective J over 50 test samples for 1D Burgers control. Baselines: PID, SAC, BC, BPPO, SL, DDPM. Metric: J (mean +/- std).",
      "source": "Appendix C.5, Table 10"
    },
    {
      "id": "wdno-D3-021",
      "claim": "Ablation: guidance importance. Compare WDNO 1D Burgers control with (lambda=120000) vs without (lambda=0) guidance term. Metric: J.",
      "source": "Appendix C.7, Table 15"
    },
    {
      "id": "wdno-D3-022",
      "claim": "Wavelet reconstruction fidelity test. Measure relative L2 reconstruction error of wavelet bases (bior1.3, bior2.4, db4, sym4) on 1D Burgers and 2D fluid training data. Metric: relative L2 error.",
      "source": "Appendix A, Table 3"
    },
    {
      "id": "wdno-D3-023",
      "claim": "Computational efficiency: wavelet vs Fourier transform. Measure total transform time on 1D Navier-Stokes training set (A100 GPU, batch 2000). Compare DWT (pytorch_wavelets) vs PyTorch 2D FFT. Metric: total transform time (seconds).",
      "source": "Appendix A, Table 4"
    },
    {
      "id": "wdno-D3-024",
      "claim": "Computational resource comparison: Parameter counts, inference time (batch 1, A100), and training time for all 1D Burgers baselines (PID, SAC, BC, BPPO, SL, DDPM, WNO, MWT, OFormer, FNO, CNN). Metrics: params, inference time, training time.",
      "source": "Appendix C.6, Table 11, Table 12, Table 13"
    },
    {
      "id": "wdno-D3-025",
      "claim": "Super-resolution computational cost scaling. Measure inference time and GPU memory for batch-5 generation at 0, 1, 2, 3 levels of super-resolution on 1D Burgers. Metrics: time, memory.",
      "source": "Appendix C.6, Table 14"
    },
    {
      "id": "wdno-D3-026",
      "claim": "WDNO training protocol (shared across all experiments). BRM and SRM trained with DDPM loss. Adam optimizer, lr=1e-4, cosine annealing, 190000 steps, batch 16. DDIM inference: 50 steps (1D) / 100 steps (2D), eta=1. Control guidance: 120000 (1D) / 100 (2D) with cosine scheduler. Training hardware: 1 A100 (1D, 2.4-2.5h) / 2xA100 (2D, 7.8-7.9h).",
      "source": "Section 3, Table 18, Table 19, Table 20, Table 12"
    }
  ],
  "D4": [
    {
      "id": "wdno-D4-001",
      "claim": "WDNO Training Pipeline (Section 3, Figure 1): Phase 1 - Train Base-Resolution Model (BRM) on full-resolution wavelet-domain data using DDPM loss (output: trained BRM; Adam, lr=1e-4, cosine annealing, 190k steps, batch 16). Phase 2 - Prepare multi-resolution dataset by downsampling data, applying wavelet transform at each resolution level, and aligning coefficient dimensions via duplication (output: multi-resolution wavelet dataset, feeds into Phase 3). Phase 3 - Train Super-Resolution Model (SRM) on paired multi-resolution wavelet data (input: Phase 2 dataset, output: trained SRM; condition on low-res coeffs + high-res params to predict high-res coeffs via DDPM loss). At inference: Phase 1 BRM generates base-resolution coefficients that feed Phase 3 SRM for iterative upsampling.",
      "source": "Section 3, Figure 1, Algorithm 1"
    },
    {
      "id": "wdno-D4-002",
      "claim": "WDNO Simulation Inference Pipeline (Section 3.1, Algorithm 1): Step 1 - Apply Discrete Wavelet Transform (DWT) to input equation parameters (IC, BC, PDE coeffs) to obtain wavelet-domain condition W_a. Step 2 - Initialize wavelet-domain trajectory as Gaussian noise W_u^{(K)}~N(0,I); iteratively denoise via DDIM for k=K..1: W_u^{(k-1)} = W_u^{(k)} - eta * epsilon_theta(W_u^{(k)}, W_a, k) + xi. Step 3 - Apply Inverse DWT to W_u^{(0)} to recover full trajectory u_{[0,T]} in original spatiotemporal domain.",
      "source": "Section 3.1, Eq. 3, Algorithm 1"
    },
    {
      "id": "wdno-D4-003",
      "claim": "WDNO Control Inference Pipeline (Section 3.1, Eq. 4-5): Step 1 - Apply DWT to input params to obtain W_a; define control objective J (state deviation at final time + energy regularization). Step 2 - Initialize noise W_f^{(K)}~N(0,I); iteratively denoise with classifier-free guidance + objective gradient for k=K..1: W_f^{(k-1)} = W_f^{(k)} - eta * (epsilon_theta(W_f^{(k)}, W_a, k) + lambda * grad_{W_f} J(W_hat_f^{(k)})) + xi, where W_hat_f is noise-free estimate of W_f^{(0)}. Step 3 - Apply IDWT to W_f^{(0)} to recover controlled trajectory.",
      "source": "Section 3.1, Eq. 4, Eq. 5"
    },
    {
      "id": "wdno-D4-004",
      "claim": "WDNO Zero-shot Super-Resolution Pipeline (Section 3.2): Step 1 - Downsample input params to base resolution NxM and apply DWT. Step 2 - BRM generates wavelet coefficients at base NxM via DDIM denoising. Step 3 - SRM iteratively upsamples: condition on current low-res coeffs + high-res params to generate coeffs at 2Nx2M; repeat until target resolution. Step 4 - Apply IDWT to obtain final trajectory at target (super-resolution) resolution.",
      "source": "Section 3.2"
    },
    {
      "id": "wdno-D4-005",
      "claim": "1D Burgers Data Generation Pipeline (Appendix F.2): Step 1 - Generate initial condition u(0,x) as superposition of 2 Gaussians with random parameters (output: initial condition, feeds Step 3). Step 2 - Generate control force f(t,x) as superposition of 8 Gaussians (a_i for i>=2 set to 0 with 50% prob) (output: control force, feeds Step 3). Step 3 - Run finite difference solver (input: Step 1 IC + Step 2 force; du/dt = -u*du/dx + nu*d^2u/dx^2 + f, Dirichlet u=0, nu=0.01, internal 120x16 spatial x 4800x16 temporal). Step 4 - Downsample solver output by factor 16 spatially and temporally to final [81,120] state + [80,120] force (input: Step 3 raw trajectory).",
      "source": "Appendix F.1, F.2"
    },
    {
      "id": "wdno-D4-006",
      "claim": "Supervised Learning (SL) Control Two-Stage Pipeline (Appendix I.3): Stage 1 - Train surrogate models (two CNN VAEs for state u and control f encoding; CNN for latent transition u_t -> u_{t+1}). Stage 2 - Optimize control f by computing gradient of objective J via backpropagation through the three surrogate networks; optimize with LBFGS (lr=0.1, 100 epochs, tol=1e-5, obj_weight=1, recon_weight=0.01).",
      "source": "Appendix I.3, Table 23"
    }
  ]
}