{
  "scope": "Software correctness and short training diagnostics only; no convergence, solution accuracy or speedup claim.",
  "paper": "https://arxiv.org/abs/2602.19475v1",
  "torch": "2.14.1+cu130",
  "device": "cpu",
  "gpu": null,
  "resume_tolerance": {
    "rtol": 1e-05,
    "atol": 1e-07,
    "cpu_requires_bitwise_identity": true
  },
  "cases": {
    "scale_allen_cahn": {
      "configuration": {
        "shape": "SpaceTime1D",
        "x_min": -1.0,
        "x_max": 1.0,
        "y_min": 0.0,
        "y_max": 1.0,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 5.0,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.001,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 1.0,
        "boundary_weight": 100.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 0.0
        },
        "boundary_sampling": {},
        "pde_expression": "",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 100,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "paper_case",
        "viscosity": 0.0001,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "expert_allen_cahn",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.45,
          "tau_alpha": 1.5,
          "gamma": 0.0001,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 1.0,
        "initial": 50.0,
        "periodic_x": 50.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 1.2245675325393677,
        "corrected_pde": 1.2245675325393677,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 1.8029226064682007,
        "corrected_pde": 1.8405935764312744,
        "correction_rms": 0.016129586845636368
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 1.6616846323013306,
        "corrected_pde": 1.6616846323013306,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 1.2245675325393677,
        "initial": 0.44496965408325195,
        "periodic_x": 0.13112178444862366,
        "total": 30.029138565063477
      },
      "final_losses": {
        "pde": 1.8405935764312744,
        "initial": 0.15128406882286072,
        "periodic_x": 0.024029580876231194,
        "total": 10.60627555847168
      }
    },
    "scale_cavity_400": {
      "configuration": {
        "shape": "Rectangle",
        "x_min": 0.0,
        "x_max": 1.0,
        "y_min": 0.0,
        "y_max": 1.0,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 0.0,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.0005,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 3.0,
        "boundary_weight": 15.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 1.0
        },
        "boundary_sampling": {},
        "pde_expression": "",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 100,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "paper_case",
        "viscosity": 0.0025,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "expert_cavity",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.06,
          "tau_alpha": 1.0,
          "gamma": 0.0025,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 3.0,
        "lid": 5.0,
        "walls": 5.0,
        "pressure_gauge": 1.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 0.08620154112577438,
        "corrected_pde": 0.08620154112577438,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 0.1357755810022354,
        "corrected_pde": 0.05667279288172722,
        "correction_rms": 0.28503870964050293
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 0.08058378845453262,
        "corrected_pde": 0.08058378845453262,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 0.08620154112577438,
        "lid": 1.4830721616744995,
        "walls": 0.6771826148033142,
        "pressure_gauge": 2.355238437652588,
        "total": 13.415117263793945
      },
      "final_losses": {
        "pde": 0.05667279288172722,
        "lid": 0.9044514894485474,
        "walls": 0.34735026955604553,
        "pressure_gauge": 1.7898681163787842,
        "total": 8.218894958496094
      }
    },
    "scale_cavity_3200": {
      "configuration": {
        "shape": "Rectangle",
        "x_min": 0.0,
        "x_max": 1.0,
        "y_min": 0.0,
        "y_max": 1.0,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 0.0,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.0005,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 3.0,
        "boundary_weight": 15.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 1.0
        },
        "boundary_sampling": {},
        "pde_expression": "",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 100,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "paper_case",
        "viscosity": 0.0003125,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "expert_cavity",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.06,
          "tau_alpha": 1.0,
          "gamma": 0.0003125,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 3.0,
        "lid": 5.0,
        "walls": 5.0,
        "pressure_gauge": 1.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 0.08605441451072693,
        "corrected_pde": 0.08605441451072693,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 0.1360645592212677,
        "corrected_pde": 0.05670095980167389,
        "correction_rms": 0.2850300371646881
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 0.08063089102506638,
        "corrected_pde": 0.08063089102506638,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 0.08605441451072693,
        "lid": 1.4830721616744995,
        "walls": 0.6771826148033142,
        "pressure_gauge": 2.355238437652588,
        "total": 13.414676666259766
      },
      "final_losses": {
        "pde": 0.05670095980167389,
        "lid": 0.9052742719650269,
        "walls": 0.34799137711524963,
        "pressure_gauge": 1.7902770042419434,
        "total": 8.22670841217041
      }
    },
    "scale_cavity_7500": {
      "configuration": {
        "shape": "Rectangle",
        "x_min": 0.0,
        "x_max": 1.0,
        "y_min": 0.0,
        "y_max": 1.0,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 0.0,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.0005,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 3.0,
        "boundary_weight": 10.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 1.0
        },
        "boundary_sampling": {},
        "pde_expression": "",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 100,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "paper_case",
        "viscosity": 0.00013333333333333334,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "expert_cavity",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.095,
          "tau_alpha": 0.5,
          "gamma": 0.00013333333333333334,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 3.0,
        "lid": 3.3333333333333335,
        "walls": 3.3333333333333335,
        "pressure_gauge": 1.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 0.08604257553815842,
        "corrected_pde": 0.08604257553815842,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 0.12244658917188644,
        "corrected_pde": 0.05592896416783333,
        "correction_rms": 0.22021278738975525
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 0.07395342737436295,
        "corrected_pde": 0.07395342737436295,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 0.08604257553815842,
        "lid": 1.4830721616744995,
        "walls": 0.6771826148033142,
        "pressure_gauge": 2.355238437652588,
        "total": 9.814214706420898
      },
      "final_losses": {
        "pde": 0.05592896416783333,
        "lid": 0.8118859529495239,
        "walls": 0.26063618063926697,
        "pressure_gauge": 1.48675537109375,
        "total": 5.229616165161133
      }
    },
    "scale_cavity_10000": {
      "configuration": {
        "shape": "Rectangle",
        "x_min": 0.0,
        "x_max": 1.0,
        "y_min": 0.0,
        "y_max": 1.0,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 0.0,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.0005,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 3.0,
        "boundary_weight": 10.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 1.0
        },
        "boundary_sampling": {},
        "pde_expression": "",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 100,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "paper_case",
        "viscosity": 0.0001,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "expert_cavity",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.095,
          "tau_alpha": 0.5,
          "gamma": 0.0001,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 3.0,
        "lid": 3.3333333333333335,
        "walls": 3.3333333333333335,
        "pressure_gauge": 1.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 0.08604037016630173,
        "corrected_pde": 0.08604037016630173,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 0.12245163321495056,
        "corrected_pde": 0.05592992529273033,
        "correction_rms": 0.22021406888961792
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 0.07395395636558533,
        "corrected_pde": 0.07395395636558533,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 0.08604037016630173,
        "lid": 1.4830721616744995,
        "walls": 0.6771826148033142,
        "pressure_gauge": 2.355238437652588,
        "total": 9.814208984375
      },
      "final_losses": {
        "pde": 0.05592992529273033,
        "lid": 0.8118957877159119,
        "walls": 0.26064059138298035,
        "pressure_gauge": 1.4867565631866455,
        "total": 5.229667663574219
      }
    },
    "scale_cavity_20000": {
      "configuration": {
        "shape": "Rectangle",
        "x_min": 0.0,
        "x_max": 1.0,
        "y_min": 0.0,
        "y_max": 1.0,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 0.0,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.0005,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 3.0,
        "boundary_weight": 10.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 1.0
        },
        "boundary_sampling": {},
        "pde_expression": "",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 100,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "paper_case",
        "viscosity": 5e-05,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "expert_cavity",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.095,
          "tau_alpha": 0.5,
          "gamma": 5e-05,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 3.0,
        "lid": 3.3333333333333335,
        "walls": 3.3333333333333335,
        "pressure_gauge": 1.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 0.08603707700967789,
        "corrected_pde": 0.08603707700967789,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 0.12245897203683853,
        "corrected_pde": 0.05593119561672211,
        "correction_rms": 0.22021602094173431
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 0.07395469397306442,
        "corrected_pde": 0.07395469397306442,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 0.08603707700967789,
        "lid": 1.4830721616744995,
        "walls": 0.6771826148033142,
        "pressure_gauge": 2.355238437652588,
        "total": 9.814199447631836
      },
      "final_losses": {
        "pde": 0.05593119561672211,
        "lid": 0.8119104504585266,
        "walls": 0.26064735651016235,
        "pressure_gauge": 1.4867568016052246,
        "total": 5.229743003845215
      }
    },
    "scale_ks": {
      "configuration": {
        "shape": "SpaceTime1D",
        "x_min": 0.0,
        "x_max": 6.283185307179586,
        "y_min": 0.0,
        "y_max": 0.4,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 6.25,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.001,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 1.0,
        "boundary_weight": 1000.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 0.0
        },
        "boundary_sampling": {},
        "pde_expression": "",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 100,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "paper_case",
        "viscosity": 0.390625,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "expert_ks",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.2,
          "tau_alpha": 1.5,
          "gamma": 0.390625,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 1.0,
        "initial": 500.0,
        "periodic_x": 500.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 2.0601840019226074,
        "corrected_pde": 2.0601840019226074,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 2.580350160598755,
        "corrected_pde": 2.367074728012085,
        "correction_rms": 0.07107918709516525
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 2.7164969444274902,
        "corrected_pde": 2.7164969444274902,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 2.0601840019226074,
        "initial": 0.43963149189949036,
        "periodic_x": 0.03530648723244667,
        "total": 239.5291748046875
      },
      "final_losses": {
        "pde": 2.367074728012085,
        "initial": 0.4072812795639038,
        "periodic_x": 0.00995238870382309,
        "total": 210.98391723632812
      }
    },
    "scale_kdv": {
      "configuration": {
        "shape": "SpaceTime1D",
        "x_min": -1.0,
        "x_max": 1.0,
        "y_min": 0.0,
        "y_max": 1.0,
        "t_min": 0.0,
        "t_max": 1.0,
        "cylinder_x": 0.2,
        "cylinder_y": 0.2,
        "cylinder_radius": 0.05,
        "fourth_order": 0.00152587890625,
        "startup_time": 0.1,
        "points": 64,
        "boundary_points": 8,
        "adaptive_sampling": {},
        "sampling": "random",
        "seed": 7,
        "source": 1.0,
        "layers": [
          16,
          16
        ],
        "iterations": 20,
        "lr": 0.001,
        "device": "cpu",
        "loss_formulation": "mean",
        "pde_weight": 1.0,
        "boundary_weight": 2000.0,
        "bc_types": {
          "left": "Dirichlet",
          "right": "Dirichlet",
          "bottom": "Dirichlet",
          "top": "Dirichlet"
        },
        "bc_values": {
          "left": 0.0,
          "right": 0.0,
          "bottom": 0.0,
          "top": 0.0
        },
        "boundary_sampling": {
          "left": {
            "points": 32,
            "method": "random",
            "seed": 1
          },
          "right": {
            "points": 32,
            "method": "random",
            "seed": 2
          },
          "bottom": {
            "points": 32,
            "method": "random",
            "seed": 3
          }
        },
        "pde_expression": "D(u,t)+c*u*D(u,x)+nu*D(D(D(u,x),x),x)",
        "bc_targets": {},
        "bc_operators": {},
        "interior_data": {},
        "boundary_data": {},
        "network_seed": 7,
        "method": "scale",
        "expert_settings": {
          "chunks": 8,
          "causal_epsilon": 1.0,
          "balance_every": 1000,
          "momentum": 0.9,
          "decay_rate": 0.9,
          "decay_steps": 2000,
          "gradient_floor": 1e-12,
          "weight_cap": 1000000.0,
          "balance_mode": "gradient",
          "ntk_points": 16,
          "windows": 1,
          "window_steps": 1000
        },
        "physics": "scalar",
        "viscosity": 0.00048399999999999995,
        "flow_residuals": [
          "u*D(u,x)+v*D(u,y)+D(p,x)-nu*laplace(u)",
          "u*D(v,x)+v*D(v,y)+D(p,y)-nu*laplace(v)",
          "D(u,x)+D(v,y)"
        ],
        "flow_boundary": {
          "left": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "right": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "bottom": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          },
          "top": {
            "u": {
              "type": "Dirichlet",
              "target": "y",
              "operator": ""
            },
            "v": {
              "type": "Dirichlet",
              "target": "0",
              "operator": ""
            }
          }
        },
        "pressure_point": [
          0.0,
          0.0
        ],
        "pressure_value": 0.0,
        "pressure_weight": 1.0,
        "pde_preset": "kdv_1d",
        "pde_forcing": "0",
        "sa_lr": 0.005,
        "sa_initial": 1.0,
        "sa_max": 1000.0,
        "sa_boundary": false,
        "lra_alpha": 0.9,
        "lra_interval": 10,
        "lra_epsilon": 1e-12,
        "lra_max": 1000000.0,
        "network_type": "fourier",
        "network_options": {
          "encoding": "gaussian",
          "features": 8,
          "scale": 0.15915494309189535
        },
        "scale_settings": {
          "tau_sc": 0.1,
          "tau_alpha": 20.0,
          "gamma": 0.022,
          "strength": 1.0,
          "fresh": true
        }
      },
      "loss_weights": {
        "pde": 1.0,
        "initial": 1000.0,
        "periodic_x": 1000.0
      },
      "steps": 20,
      "checkpoint_resume_identical": true,
      "checkpoint_resume_close": true,
      "parameter_max_abs_difference": 0.0,
      "initial_diagnostics": {
        "step": 1,
        "physical_pde": 0.4001043140888214,
        "corrected_pde": 0.4001043140888214,
        "correction_rms": 0.0
      },
      "final_diagnostics": {
        "step": 20,
        "physical_pde": 0.5523208379745483,
        "corrected_pde": 0.5888338088989258,
        "correction_rms": 0.031302016228437424
      },
      "uncorrected_same_batch_control": {
        "step": 20,
        "physical_pde": 0.5607329607009888,
        "corrected_pde": 0.5607329607009888,
        "correction_rms": 0.0
      },
      "initial_losses": {
        "pde": 0.4001043140888214,
        "initial": 0.9360496997833252,
        "periodic_x": 0.1073775365948677,
        "total": 1043.8272705078125
      },
      "final_losses": {
        "pde": 0.5888338088989258,
        "initial": 0.5283352732658386,
        "periodic_x": 0.04292825981974602,
        "total": 571.8523559570312
      }
    }
  }
}